commit 6c74872af07f02ef9d8c0882959368d8b3d2661c
parent 0a0cbb5730039680562f1a6c1f0b7a0c712536d8
Author: miksa <milutin@popovic.xyz>
Date: Tue, 27 Jul 2021 12:56:44 +0200
checkpoint
Diffstat:
13 files changed, 1415 insertions(+), 5511 deletions(-)
diff --git a/src/thesis/12 b/src/thesis/12
@@ -0,0 +1,49 @@
+\documentclass[12pt]{article}
+
+%-------------------- BACKHAND ---------------------
+
+\input{back/packages}
+
+\begin{document}
+
+\input{back/title}
+
+\newpage
+
+\tableofcontents
+
+\newpage
+
+\input{back/abstract}
+
+%------------------- INTRO -------------------------
+
+\input{chapters/intro}
+
+%----------------- MAIN SECTION --------------------
+
+\input{chapters/main_sec}
+
+%\input{chapters/basics}
+
+\input{chapters/finitencg}
+
+%\input{chapters/realncg}
+
+%\input{chapters/heatkernel}
+%
+%\input{chapters/twopointspace}
+%
+%\input{chapters/electroncg}
+
+%------------------ OUTRO -------------------------
+
+\input{chapters/conclusion}
+
+\input{chapters/acknowledgment}
+
+%------------------- BACKHAND ---------------------
+
+\input{back/refs}
+
+\end{document}
diff --git a/src/thesis/back/title.tex b/src/thesis/back/title.tex
@@ -34,9 +34,12 @@
{ \fontsize{10}{0} \selectfont Vienna, July 2021}\\
-\vspace*{3.5cm}
+\vspace*{3.4cm}
\begin{tabular}{p{9cm}p{11.25cm}}
+ \fontsize{10}{0} \selectfont
+ student ID number:\vspace*{0.3cm}& \fontsize{10}{0} \selectfont 11807930\\
+
\fontsize{10}{0} \selectfont degree programme code as it appears on / &
\fontsize{10}{0} \selectfont A 033 676 \\
@@ -46,11 +49,11 @@
\fontsize{10}{0} \selectfont degree
programme as it appears on / & \fontsize{10}{0} \selectfont Physics \\
- \fontsize{10}{0} \selectfont the student record sheet:\vspace*{0.4cm} &
+ \fontsize{10}{0} \selectfont the student record sheet:\vspace*{0.3cm} &
\fontsize{10}{0} \selectfont \\
\fontsize{10}{0} \selectfont
- Supervisor:\vspace*{0.4cm}& \fontsize{10}{0} \selectfont Lisa Glaser, PhD\\
+ Supervisor:\vspace*{0.3cm}& \fontsize{10}{0} \selectfont Lisa Glaser, PhD\\
\end{tabular}
\end{center}
\end{titlepage}
diff --git a/src/thesis/chapters/basics.tex b/src/thesis/chapters/basics.tex
@@ -8,15 +8,15 @@ introduce the first ingredient of a spectral triple, an unital $C^*$ algebra.
\begin{enumerate}
\item
$A \times A \rightarrow A$,
- $(a, b)\ \mapsto \ a\cdot b$,
- \item with an identity element
+ $(a,\ b)\ \mapsto \ a\cdot b$,
+ \item with an identity element:
$1a = a1 =a$.
\end{enumerate}
Extending the definition, a $*$-algebra is an algebra $A$ with a \textit{conjugate linear map (involution)} $*:A\ \rightarrow A$,
$\forall a, b \in A$ satisfying:
\begin{enumerate}
\item
- $(ab)^* = b^*a^*$,
+ $(a\ b)^* = b^*a^*$,
\item
$(a^*)^* = a$.
\end{enumerate}
@@ -26,14 +26,14 @@ In the following all unital algebras are referred to as algebras.
\subsubsection{Finite Discrete Space}
Let us consider an example of an $*$-algebra of continuous functions $C(X)$
on a discrete topological space $X$ with $N$ points. Functions of a
-continuous $*$-algebra $C(X)$ assign values to $\mathbb{C}$, thus $f, g \in
+continuous $*$-algebra $C(X)$ assign values to $\mathbb{C}$, thus $f,\ g \in
C(X)$, $\lambda \in \mathbb{C}$ and $x \in X$ they provide the following structure:
\begin{itemize}
\item \textit{pointwise linear} \\
$(f + g)(x) = f(x) + g(x)$,\\
- $(\lambda f)(x) = \lambda (f(x)),$
+ $(\lambda\ f)(x) = \lambda (f(x)),$
\item \textit{pointwise multiplication} \\
- $fg(x) = f(x)g(x)$,
+ $f\ g\ (x) = f(x)g(x)$,
\item \textit{pointwise involution} \\
$f^*(x) = \overline{f(x)}.$
\end{itemize}
@@ -60,9 +60,9 @@ Note that the pullback doesn't map points back, but maps functions on an $*$-alg
The pullback, in literature often called a $*$-homomorphism or a $*$-algebra map under
pointwise product has the following properties
\begin{itemize}
- \item $\phi ^*(fg) = \phi ^*(f) \phi ^*(g)$,
+ \item $\phi ^*(f\ g) = \phi ^*(f)\ \phi ^*(g)$,
\item $\phi ^*(\overline{f}) = \overline{\phi ^*(f)}$,
- \item $\phi ^*(\lambda f + g) = \lambda \phi ^*(f) + \phi ^*(g)$.
+ \item $\phi ^*(\lambda\ f + g) = \lambda\ \phi ^*(f) + \phi ^*(g)$.
\end{itemize}
%------------ Exercise
The map $\phi :X_1\ \rightarrow \ X_2$ is an injective (surjective) map,
@@ -102,8 +102,9 @@ $(\cdot,\cdot)\rightarrow \mathbb{C}$. We denote $L(H)$ as the $*$-algebra of o
equipped with a product given by composition and involution of the adjoint, $T \mapsto T^*$.
Then $L(H)$ is a \textit{normed vector space} with
\begin{align}
- &\|T\|^2 = \text{sup}_{h \in H}\{(Th,Th): (h,h) \leq 1\} \hspace{0.1\textwidth} T \in L(H) \\
- &\|T\| = \text{sup}\{\sqrt{\lambda}: \lambda \text{ eigenvalue of } T\}
+ \|T\|^2 &= \sup_{h \in H}\big\{(T\ h,\ T\ h): (h,\ h) \leq 1\big|\ T
+ \in L(H)\big \},\\
+ \|T\| &= \sup\big\{\sqrt{\lambda}:\; \lambda \text{ eigenvalue of } T\big\}.
\end{align}
This allows us to define representations of $*$-algebras.
\begin{definition}
@@ -150,7 +151,8 @@ triple and a space.
Given a representation $(H, \pi)$ of a $*$-algebra $A$, the \textbf{commutant} $\pi (A)'$ of $\pi (A)$ is defined as a set
of operators in $L(H)$ that commute with all $\pi (a)$
\begin{align}
- \pi (A)' = \{T \in L(H):\pi (a)T = T\pi (a) \;\;\; \forall a\in A\}
+ \pi (A)' = \big\{T \in L(H):\ \pi(a)\ T = T\ \pi(a) \;\; \forall a\in
+ A\big\}
\end{align}
The commutant $\pi (A)'$ is also a $*$-algebra, because it has unital,
associative and involutive properties.
@@ -182,7 +184,7 @@ triple and a space.
\begin{align}
\pi (a) &= \mathbbm{1}_n^*\ \tilde{\pi}((a_{ij})), \\
\mathbbm{1}_n &= \tilde{\pi}((a_{ij})) = \pi (a_{ij})
- \Rightarrow a_{ij} = a\mathbbm{1}_n.
+ \Rightarrow a_{ij} = a\ \mathbbm{1}_n.
\end{align}
%------------- EXERCISE
@@ -207,1373 +209,347 @@ generalized notion of isomorphisms between matrix algebras (\textit{Morita
Equivalence})
\subsubsection{Algebraic Modules}
+An important notion for Morita Equivalence are algebraic modules, later
+extended with Hilbert bimodules.
\begin{definition}
Let $A$, $B$ be algebras (need not be matrix algebras)
\begin{enumerate}
- \item \textit{left} A-module is a vector space $E$, that carries a left representation of $A$,
- that is $\exists$ a bilinear map $\gamma: A \times E \rightarrow E$ with
+ \item \textit{left} A-module is a vector space $E$, that carries a left
+ representation of $A$, that is $\exists$ a bilinear map $\gamma: A
+ \times E \rightarrow E$ with
\begin{align}
- (a_1a_2)\cdot e = a_1 \cdot (a_2 \cdot e);\;\;\; a_1, a_2 \in A, e \in E
+ (a_1\ a_2)\cdot e = a_1 \cdot (a_2 \cdot e);\;\;\; a_1, a_2 \in
+ A, e \in E.
\end{align}
- \item \textit{right} B-module is a vector space $F$, that carries a right representation of $A$,
- that is $\exists$ a bilinear map $\gamma: F \times B \rightarrow F$ with
+ \item \textit{right} B-module is a vector space $F$, that carries a
+ right representation of $A$, that is there exists a bilinear map
+ $\gamma: F \times B \rightarrow F$ with
\begin{align}
- f \cdot (b_1b_2)= (f \cdot b_1) \cdot b_2;\;\;\; b_1, b_2 \in B, f \in F
+ f \cdot (b_1\ b_2)= (f \cdot b_1) \cdot b_2;\;\;\; b_1, b_2 \in B, f \in F
\end{align}
- \item \textit{left} A-module and \textit{right} B-module is a \textit{bimodule}, a vector space $E$
- satisfying
+ \item \textit{left} A-module and \textit{right} B-module is a
+ \textit{bimodule}, a vector space $E$ satisfying
\begin{align}
a \cdot (e \cdot b)= (a \cdot e) \cdot b;\;\;\; a \in A, b \in B, e \in E
\end{align}
\end{enumerate}
\end{definition}
-Notion of A-\textbf{module homomorphism} as linear map $\phi: E\rightarrow F$ which respects the
-representation of A, e.g. for left module.
+An $A$-\textbf{module homomorphism} as linear map $\phi: E\rightarrow F$ which respects the
+representation of A, e.g.\ for left module.
\begin{align}
- \phi (ae) = a \phi (e); \;\;\; a \in A, e \in E.
+ \phi (a\ e) = a \phi (e); \;\;\; a \in A, e \in E.
\end{align}
-Remark on the notation
+We will use the notation
\begin{itemize}
- \item ${}_A E$ left $A$-module $E$;
- \item ${}_A E_B$ right $B$-module $F$;
- \item ${}_A E_B$ $A$-$B$-bimodule $E$;
+ \item ${}_A E$, for left $A$-module $E$;
+ \item ${}_A E_B$, for right $B$-module $F$;
+ \item ${}_A E_B$, for $A$-$B$-bimodule $E$, simply bimodule.
\end{itemize}
+%------------------- EXERCISE
+From a simple observation, we see that an arbitrary representation $\pi : A
+\rightarrow L(H)$ of a $*$-algebra A, turns H into a left module ${}_A H$. If
+$_A H$ than $(a_1\ a_2) h = a_1 (a_2\ h)$ for $a_1, a_2 \in A$ and $h \in H$. We
+take the representation of an $a \in A$, $\pi (a)$, and write
+\begin{align}
+ \big(\pi(a_1)\ \pi(a_2)\big)h = \pi(a_1)\big(\pi(a_2)\ h\big) =
+ \big(T_1\ T_2\big) h = T_1 \big(T_2\ h\big)
+\end{align}
+For $T_1, T_2 \in L(H)$, which operate naturally from the left on $h$.
+
+%------------------- EXERCISE
+%------------------- EXERCISE
+
+Furthermore notice that that an $*$-algebra $A$ is a bimodule ${}_A A_A$ with
+itself, given by the map
+\begin{align}
+ \gamma: A\times A\times A \rightarrow A,
+\end{align}
+which is the inner product of a $*$-algebra.
+%------------------- EXERCISE
+
+\subsubsection{Balanced Tensor Product and Hilbert Bimodules}
+
+\begin{definition}
+ Let $A$ be an algebra, $E$ be a \textit{right} $A$-module and $F$ be a
+ \textit{left} $A$-module. The \textit{balanced tensor product} of $E$ and
+ $F$ forms a $A$-bimodule.
+ \begin{align}
+ E \otimes _A F := E \otimes F / \left\{\sum _i e_i a_i \otimes f_i -
+ e_i \otimes a_i f_i : \;\;\; a_i \in A,\ e_i \in E,\ f_i \in F
+ \right\}.
+ \end{align}
+\end{definition}
+The $/$ denotes the quotient space. By that the operation $\otimes _A$ takes
+two left/right modules and makes a bimodule with the help the tensor product of
+the two modules and the quotient space that takes out all the elements from the
+tensor product that don't preserver the left/right representation and that are
+duplicates.
+\begin{definition}
+ Let $A$, $B$ be \textit{matrix algebras}. The \textit{Hilbert bimodule} for
+ $(A, B)$ is given by an $A$-$B$-bimodue $E$ and by an $B$-valued
+ \textit{inner product} $\langle \cdot,\cdot\rangle_E: E\times E \rightarrow
+ B$, which satisfies the following conditions for $e, e_1, e_2 \in
+ E,\ a \in A$ and $b \in B$
+\begin{align}
+ \langle e_1,\ a\cdot e_2\rangle_E &= \langle a^*\cdot e_1,\ e_2\rangle_E
+ \;\;\;\; & \text{sesquilinear in $A$},\\
+ \langle e_1,\ e_2 \cdot b\rangle_E
+ &= \langle e_1,\ e_2\rangle_E b \;\;\;\; & \text{scalar in $B$},\\
+ \langle e_1,\ e_2\rangle_E &= \langle e_2,\ e_1\rangle^*_E \;\;\;\; &
+ \text{hermitian}, \\
+ \langle e,\ e\rangle_E &\ge 0 \;\;\;\; & \text{equality
+ holds iff $e=0$}.
+\end{align}
+We denote $KK_f(A,\ B)$ as the set of all \textit{Hilbert bimodules} of $(A,\ B)$.
+\end{definition}
+%-------------- EXERCISE
-\begin{MyExercise}
- \textbf{
- Check that a representation of $\pi : A \rightarrow L(H)$ of a $*$-algebra A turns H into a
- left module ${}_A H$.
-}\newline
+And indeed the Hilbert bimodule extension takes a representation $\pi:\ A \
+\rightarrow L(H)$ of a matrix algebra $A$ and turns $H$ into a Hilbert bimodule for
+$(A, \mathbb{C})$, because the representation of $a \in A$, $\pi(a)=T \in L(H)$ fulfills
+the conditions of the $\mathbb{C}$-valued inner product for $h_1, h_2 \in H$
+\begin{itemize}
+ \item $\langle h_1,\ \pi(a)\ h_2\rangle _\mathbb{C} = \langle h_1,\ T\ h_2\rangle _\mathbb{C} =
+ \langle T^* h_1, h_2\rangle _\mathbb{C}$, $T^*$ given by the adjoint,
+ \item $\langle h_1,\ h_2\ \pi(a)\rangle _\mathbb{C} = \langle h_1,\ h_2\
+ T\rangle _\mathbb{C} = \langle h_1,\ h_2\rangle _\mathbb{C}$ , $T$ acts
+ from the left,
+ \item $\langle h_1,\ h_2\rangle _\mathbb{C}^* = \langle h_2,\ h_1\rangle _\mathbb{C}$, hermitian because of the
+ $\mathbb{C}$-valued inner product
+ \item $\langle h_1,\ h_2\rangle \ge 0$, $\mathbb{C}$-valued inner product.
+\end{itemize}
+%-------------- EXERCISE
- Not quite sure but \\
- $a \in A$, $h_1, h_2 \in H$, we know $\pi (a) = T \in L(H)$ than
+%-------------- EXERCISE
+Take again the $A-A$ bimodule given by an $*$-algebra $A$, it is in
+$KK_f(A,\ A)$. This becomes clear by looking at the following inner product
+ $\langle \cdot,\cdot\rangle_A:A \times A \rightarrow A$:
\begin{align}
- \langle \pi (a) h_1, \pi (a) h_2\rangle = \langle T h_1, T h_2\rangle = \langle T^*T h_1, h_2\rangle = \langle h_1, h_2\rangle
+ \langle a,\ a\rangle_A = a^*a' \;\;\;\; a,a'\in A. \label{eq:inner-product}
\end{align}
- Or maybe this \\
- If $_A H$ than $(a_1a_2) h = a_1 (a_2 h)$ for $a_1, a_2 \in A$ and $h \in H$.\\
- Then we take the representation of an $a \in A$, $\pi (a)$:
+ Simply checking the conditions in $\langle \cdot, \cdot\rangle _A$ for
+ $a, a_1, a_2 \in A$
\begin{align}
- (\pi(a_1)\pi(a_2))h = \pi(a_1)(\pi(a_2) h) = (T_1T_2) h = T_1 (T_2 h)
+ \langle a_1,\ a\cdot a_2\rangle _A &= a^* a\cdot a_2 =
+ (a^*a_1)^*\ a_2 = \langle a^*\ a_1,\ a_2\rangle, \\
+ \langle a_1,\ a_2 \cdot a\rangle _A &= a^*_1\ (a_2\cdot a) =
+ (a^*a_2)\cdot a = \langle a_1,\ a_2\rangle _A\ a,\\
+ \langle a_1,\ a_2\rangle _A^* &= (a_1^*\ a_2)^* = a_2^*\
+ (a_1^*)^* = a_2^*\ a_1 = \langle a_2,\ a_1\rangle.
\end{align}
- For $T_1, T_2 \in L(H)$, which operate naturally from the left on $h$.
-\end{MyExercise}
-\begin{MyExercise}
- \textbf{
- Show that $A$ is a bimodule ${}_A A_A$ with itself.
-}\newline
+%-------------- EXERCISE
- $\gamma: A\times A\times A \rightarrow A$ which is given by the inner product of the $*$-algebra.
-\end{MyExercise}
+%-------------- EXAMPLE
+As an exemplar for overview consider a $*$ homomorphism between two matrix
+algebras $\phi:A\rightarrow B$, we can construct a Hilbert bimodule
+$E_{\phi} \in KK_f(A, B)$ in the following way. We let $E_{\phi}$ be $B$ in
+as an vector space and an inner product from above in equation
+\eqref{eq:inner-product}, with $A$ acting on the left with $\phi$.
+\begin{align}
+ a\cdot b = \phi(a)\ b
+\end{align}
+for $a\in A, b\in E_{\phi}$.
+%-------------- EXAMPLE
+
+\subsubsection{Kasparov Product and Morita Equivalence}
+\begin{definition}
+ Let $E \in KK_f(A, B)$ and $F \in KK_F(B, D)$ the \textit{Kasparov product} is defined as
+ with the balanced tensor product
+ \begin{align}
+ F \circ E := E \otimes _B F.
+ \end{align}
+ Then $F\circ E \in KK_f(A,D)$ is equipped with a $D$-valued inner product
+ \begin{align}
+ \langle e_1 \otimes f_1,\ e_2 \otimes f_2\rangle _{E\otimes _B F} =
+ \langle f_1,\langle e_1,\ e_2\rangle _E f_2\rangle _F
+ \end{align}
+\end{definition}
+
+%-------------- EXERCISE
+The Kasparov product for $*$-algebra homomorphism $\phi: A \rightarrow B$ and
+$\psi: B \rightarrow C$ are isomorphisms in the sense of
+\begin{align}
+ E_{\psi} \circ E_{\phi}\ \equiv\ E_{\phi} \otimes _B E_{\psi}\
+ \simeq\
+ E_{\psi \circ \phi} \in KK_f(A,C).
+\end{align}
+
+In the direct computation for elements $a \in A$, $b\in B$, and $c\in C$ which
+is $\psi \circ \phi$ gives us
+\begin{align}
+a \cdot b \cdot c = \psi(\phi (a) \cdot b) \cdot c
+\end{align}
+An interesting case arises when looking at $E_{\text{id}_A} \simeq A \in KK_f(A,A)$ for
+$\text{id}_A: A \rightarrow A$. This is obvious when we let $E_{\phi}$ be $A$
+with a natural right representation. It follows that $E_{\phi}\simeq A$, thus
+an inner product, acting from the left on $A$ for $\phi$, $a', a\in A$ reads
+\begin{align}
+ a'\ a = (\phi(a')\ a) \in A,
+\end{align}
+which is satisfied by $\phi = \text{id}_A$
+
+\begin{definition}
+ Let $A$, $B$ be \textit{matrix algebras}. They are called \textit{Morita equivalent} if there
+ exists an $E \in KK_f(A, B)$ and an $F \in KK_f(B, A)$ such that
+ \begin{align}
+ E \otimes _B F \simeq A \;\;\; \text{and} \;\;\; F \otimes _A E \simeq
+ B,
+ \end{align}
+ where $\simeq$ denotes the isomorphism between Hilbert bimodules and note
+ that $A$ or $B$ is a bimodule by itself.
+\end{definition}
+
+\end{align}
+The modules $E$ and $F$ are each others inverse in regards to the Kasparov
+Product, because we land in the same space as we started. To clarify, in
+the definition we have $E \in KK_f(A, B)$. We start from $A$ and $E \otimes
+_B F$, which lands in $A$. Oppositely we have $F \in KK_f(B, D)$ we start
+from $B$ and $F \otimes _A E$, which lands in $B$.
+
+
+%------------- EXERCISE
+By definition $E \otimes _B F$ is a $A-D$ bimodule. Since
+\begin{align}
+ E \otimes _B F = E \otimes F / \bigg\{\sum_i\ e_i\ b_i \otimes f_i - e_i
+ \otimes b_i\ f_i\ \big|\;\; e_i \in E_i,\ b_i \in B,\ f_i \in F\bigg\},
+\end{align}
+the last part takes out all tensor product elements of $E$ and $F$ that don't
+preserver the left/right representation and that are duplicates.
+
+Additionally $\langle \cdot,\cdot\rangle _{E\oplus _B F}$ defines a $D$ valued
+inner product, as $\langle e_1,\ e_2\rangle _E \in B$ and $\langle f_1,\ f_2\rangle _F \in C$ by
+definition. So for $\langle e_1,\ e_2\rangle _E =b$ we have
+\begin{align}
+ \langle e_1 \otimes f_1,\ e_2 \otimes f_2\rangle _{E\otimes _B F} = \langle
+ f_1,\ \langle e_1,\ e_2\rangle _E\ f_2\rangle _F = \langle f_1,\ b\ f_2\rangle _F \in C
+\end{align}
+%------------- EXERCISE
+%------------- EXAMPLE
+Picking up the example of $(A, A)$, the Hilbert bimodule $A$, we can
+consider an $E \in KK_f(A,B)$ for
+\begin{align}
+ E \circ A = A\oplus _A E \simeq E.
+\end{align}
+We conclude, that $_A A_A$ is the identity element in the Kasparov product (up
+to isomorphism).
+%------------- EXAMPLE
+%------------- EXAMPLE
+Let us examine another example for $E = \mathbb{C}^n$, which is a
+$(M_n(\mathbb{C}), \mathbb{C})$ Hilbert bimodule with the standard $\mathbb{C}$
+inner product. Further let $F = \mathbb{C}^n$, which is a $(\mathbb{C},
+M_n(\mathbb{C}))$ Hilbert bimodule by right matrix multiplication with
+$M_n(\mathbb{C})$ valued inner product, we can write
+ \begin{align}
+ \langle v_1, v_2\rangle =\bar{v_1}v_2^t \;\; \in M_n(\mathbb{C}).
+ \end{align}
+If we take the Kasparov product of $E$ and $F$
+ \begin{align}
+ F\circ E\ &=\ E\otimes _{\mathbb{C}}F\ \;\;\;\;\;\; \simeq \
+ M_n(\mathbb{C}),\\
+ E\circ F\ &=\ F\otimes _{M_n(\mathbb{C})}E\ \simeq\ \mathbb{C},
+ \end{align}
+we see that $M_n(\mathbb{C})$ and $\mathbb{C}$ are Morita equivalent!
+%------------- EXAMPLE
+
+\begin{theorem}
+ Two matrix algebras are Morita Equivalent if, and only if their their structure spaces
+ are isomorphic as discreet spaces (have the same cardinality / same number
+ of elements).
+\end{theorem}
+\begin{proof}
+ Let $A$, $B$ be \textit{Morita equivalent}. Then there exist $_A E_B$ and $_B F_A$ with
+ \begin{align}
+ E \otimes _B F \simeq A \;\;\; \text{and} \;\;\; F \otimes _A E \simeq
+ B.
+ \end{align}
+ Also consider $[(\pi _B, H)] \in \hat{B}$. We can construct a
+ representation of $A$, which reads
+ \begin{align}
+ \pi _A \rightarrow L(E \otimes _B H)\;\;\; \text{with} \;\;\; \pi _A(a)
+ (e \otimes v) = a e \otimes w
+ \end{align}
+ Vice versa, we have $[(\pi _A, W)] \in \hat{A}$ we can construct $\pi _B$
+ as
+ \begin{align}
+ \pi _B: B \rightarrow L(F \otimes _A W) \;\;\; \text{and}\;\;\; \pi
+ _B(b) (f\otimes w) = bf\otimes w.
+ \end{align}
+ Now we need to show that the representation $\pi _A$ is irreducible if and
+ only if $\pi _B$ is irreducible. For $(\pi _B, H)$ to be irreducible, we
+ need $H \neq \emptyset$ and only $\emptyset$ or $H$ to be invariant under
+ the Action of $B$ on $H$. Than $E\otimes _B H$ and $E\otimes _B H \simeq A$
+ cannot be empty, because $E$ preserves left representation of $A$.
+
+ Lastly we need to check if the association of the class $[\pi _A]$ to $[\pi
+ _B]$ is independent of the choice of representatives $\pi _A$ and $\pi _B$.
+ The important thing is that $[\pi _A] \in \hat{A}$ respectively $[\pi _B] \in
+ \hat{B}$, hence any choice of representation is irreducible, because the
+ structure space denotes all unitary equivalence classes of irreducible
+ representations.
+
+ Note that the statements $E \simeq H$ and $F \simeq W$ are not particularly
+ true, since all infinite dimensional Hilbert spaces are isomorphic. Here
+ we are looking at finite dimensional Hilbert spaces. Another thing to keep
+ in mind, is that for $[\pi _B, H] \in \hat{B}$ and looking at algebraic
+ bimodules, we know that $H$ is a bimodule of $B$, hence $E \otimes _B
+ H\simeq A$, and for $[\pi _A, W]$, which is the same.
+ Finally we can conclude, that these maps are each others inverses, thus
+ $\hat{A} \simeq \hat{B}$.
+\end{proof}
+
+\begin{lemma}
+ The matrix algebra $M_n(\mathbb{C})$ has a unique irreducible
+ representation (up to isomorphism) given by the defining representation on
+ $\mathbb{C}^n$.
+\end{lemma}
+\begin{proof}
+ We know $\mathbb{C}^n$ is a irreducible representation of $A=
+ M_n(\mathbb{C})$. Let $H$ be irreducible and of dimension $k$, then we
+ define a map
+ \begin{align}
+ \phi : A\oplus...\oplus A &\rightarrow H^* \\
+ (a_1,...,a_k)&\mapsto e^1\circ a_1^t+...+e^k\circ a_k^t,
+ \end{align}
+where $\{e^1,...,e^k\}$ is the basis of the dual space $H^*$ and
+$(\circ)$ being the pre-composition of elements in $H^*$ and $A$ acting on $H$.
+This forms a morphism of $M_n(\mathbb{C})$ modules, provided a matrix $a \in A$
+acts on $H^*$ with $v\mapsto v\circ a^t$ ($v\in H^*$). Furthermore this
+morphism is surjective, thus making the pullback $\phi ^*:H\mapsto (A^k)^*$
+injective. Now identify $(A^k)^*$ with $A^k$ as a $A$-module and note that
+$A=M_n(\mathbb{C}) \simeq \oplus ^n \mathbb{C}^n$ as a n A module. It follows
+that $H$ is a submodule of $A^k \simeq \oplus ^{nk}\mathbb{C}$. By
+irreducibility $H \simeq \mathbb{C}$.
+\end{proof}
+
+%---------------- EXAMPLE
+Let us look at an examples for two matrix algebras $A$, and $B$.
+\begin{align}
+ A = \bigoplus ^N_{i=1} M_{n_i}(\mathbb{C}), \;\;\;
+ B = \bigoplus ^M_{j=1} M_{m_j}(\mathbb{C}).
+\end{align}
+Let $\hat{A} \simeq \hat{B}$, this implies $N=M$. Further define $E$ with $A$
+acting by block-diagonal matrices on the first tensor and B acting in the same
+manner on the second tensor. Define $F$ vice versa, ultimately reading
+\begin{align}
+ E:= \bigoplus _{i=1}^N \mathbb{C}^{n_i} \otimes \mathbb{C}^{m_i}, \;\;\;
+ F:= \bigoplus _{i=1}^N \mathbb{C}^{m_i} \otimes \mathbb{C}^{n_i}.
+\end{align}
+When we calculate the Kasparov product we get the following
+\begin{align}
+ E \otimes _B F &\simeq \bigoplus _{i=1}^N (\mathbb{C}^{n_i}\otimes\mathbb{C}^{m_i})
+ \otimes _{M_{m_i}(\mathbb{C})} (\mathbb{C}^{m_i}\otimes\mathbb{C}^{n_i}) \\
+ &\simeq \bigoplus _{i=1}^N \mathbb{C}^{n_i}\otimes
+ \left(\mathbb{C}^{m_i}\otimes _{M_{m_i}(\mathbb{C})}\mathbb{C}^{m_i}\right)
+ \oplus \mathbb{C}^{n_i} \\
+ &\simeq \bigoplus _{i=1}^N
+ \mathbb{C}^{m_i}\otimes\mathbb{C}^{n_i} \simeq A.
+\end{align}
+On the other hand we get
+\begin{align}
+ F \otimes _A E \simeq B.
+\end{align}
+%---------------- EXAMPLE
-%\subsubsection{Balanced Tensor Product and Hilbert Bimodules}
-%
-%\begin{definition}
-% Let $A$ be an algebra, $E$ be a \textit{right} $A$-module and $F$ be a \textit{left} $A$-module.
-% The \textit{balanced tensor product} of $E$ and $F$ forms a $A$-bimodule.
-% \begin{align}
-% E \otimes _A F := E \otimes F / \left\{\sum _i e_i a_i \otimes f_i - e_i \otimes a_i f_i : \;\;\;
-% a_i \in A,\ e_i \in E,\ f_i \in F \right\}
-% \end{align}
-%\end{definition}
-%Note $/$ denotes the quotient space. So $\otimes _A$ takes two left/right modules and makes a
-%bimodule with the help the tensor product of the two modules and the quotient space that takes
-%out all the elements from the tensor product that dont preserver the left/right representation and that
-%are duplicates.
-%\begin{definition}
-% Let $A$, $B$ be \textit{matrix algebras}. The \textit{Hilbert bimodule} for $(A, B)$ is given by
-% \begin{itemize}
-% \item $E$, an $A$-$B$-bimodue $E$ and by
-% \item an $B$-valued \textit{inner product} $\langle \cdot,\cdot\rangle_E: E\times E \rightarrow B$
-% \end{itemize}
-%$\langle \cdot,\cdot\rangle_E$ needs to satisfy the following for $e, e_1, e_2 \in E,\ a \in A$ and $b \in B$.
-%\begin{align}
-% \langle e_1, a\cdot e_2\rangle_E &= \langle a^*\cdot e_1, e_2\rangle_E \;\;\;\; & \text{sesquilinear in $A$}\\
-% \langle e_1, e_2 \cdot b\rangle_E &= \langle e_1, e_2\rangle_E b \;\;\;\; & \text{scalar in $B$} \\
-% \langle e_1, e_2\rangle_E &= \langle e_2,e_1\rangle^*_E \;\;\;\; & \text{hermitian} \\
-% \langle e, e\rangle_E &\ge 0 \;\;\;\; & \text{equality holds iff $e=0$}
-%\end{align}
-%
-%\end{definition}
-%
-%We denote $KK_f(A,B)$ the set of all \textit{Hilbert bimodules} of $(A,B)$.
-%
-%%\begin{MyExercise}
-%% \textbf{
-%% Check that a representation $\pi:\ A \ \rightarrow L(H)$ of a matrix algebra $A$ turns $H$ into
-%% a Hilbert bimodule for $(A, \mathbb{C})$.
-%% \label{ex: bimodule}
-%%}\newline
-%%
-%%
-%% We check if the representation of $a \in A$, $\pi(a)=T \in L(H)$ fulfills
-%% the conditions on the $\mathbb{C}$-valued inner product for $h_1, h_2 \in H$:
-%% \begin{itemize}
-%% \item $\langle h_1, \pi(a) h)2\rangle _\mathbb{C} = \langle h_1, T h_2\rangle _\mathbb{C} =
-%% \langle T^* h_1, h_2\rangle _\mathbb{C}$, $T^*$ given by the adjoint
-%% \item $\langle h_1, h_2 \pi(a)\rangle _\mathbb{C} = \langle h_1, h_2 T\rangle _\mathbb{C} = \langle h_1, h_2\rangle _\mathbb{C}$, $T$ acts from the left
-%% \item $\langle h_1, h_2\rangle _\mathbb{C}^* = \langle h_2,h_1\rangle _\mathbb{C}$, hermitian because of the
-%% $\mathbb{C}$-valued inner product
-%% \item $\langle h_1, h_2\rangle \ge 0$, $\mathbb{C}$-valued inner product.
-%% \end{itemize}
-%%\end{MyExercise}
-%
-%%\begin{MyExercise}
-%% \textbf{
-%% Show that the $A-A$ bimodule given by $A$ is in $KK_f(A,A)$ by taking the following inner product
-%% $\langle \cdot,\cdot\rangle_A:A \times A \rightarrow A$:
-%% \begin{align}
-%% \langle a, a\rangle_A = a^*a' \;\;\;\; a,a'\in A
-%% \end{align}
-%% \label{exercise: inner-product}
-%%}\newline
-%%
-%%
-%% We check again the conditions on $\langle \cdot, \cdot\rangle _A$, let $a, a_1, a_2 \in A$:
-%% \begin{itemize}
-%% \item $\langle a_1, a\cdot a_2\rangle _A = a^*\ a\cdot a_2 = (a^*a_1)^* a_2 = \langle a^*a_1, a_2\rangle $
-%% \item $\langle a_1, a_2 \cdot a\rangle _A = a^*_1 (a_2\cdot a) = (a^*a_2)\cdot a = \langle a_1, a_2\rangle _A a$
-%% \item $\langle a_1, a_2\rangle _A^* = (a_1^* a_2)^* = a_2^*(a_1^*)^* = a_2^* a_1 = \langle a_2, a_1\rangle $
-%% \end{itemize}
-%%\end{MyExercise}
-%
-%\begin{example}
-% Consider a $*$ homomorphism between two matrix algebras $\phi:A\rightarrow B$.
-% From it we can construct a Hilbert bimodule $E_{\phi} \in KK_f(A, B)$ in the following way.
-% We let $E_{\phi}$ be $B$ in the vector space sense and an inner product from the above
-% Exercise \ref{exercise: inner-product}, with $A$ acting on the left with $\phi$.
-% \begin{align}
-% a\cdot b = \phi(a)b \;\;\;\; a\in A, b\in E_{\phi}
-% \end{align}
-%\end{example}
-%
-%
-%\subsubsection{Kasparov Product and Morita Equivalence}
-%\begin{definition}
-% Let $E \in KK_f(A, B)$ and $F \in KK_F(B, D)$ the \textit{Kasparov product} is defined as
-% with the balanced tensor product
-% \begin{align}
-% F \circ E := E \otimes _B F
-% \end{align}
-% Such that $F\circ E \in KK_f(A,D)$ with a $D$-valued inner product.
-% \begin{align}
-% \langle e_1 \otimes f_1, e_2 \otimes f_2\rangle _{E\otimes _B F} = \langle f_1,\langle e_1, e_2\rangle _E f_2\rangle _F
-% \end{align}
-%\end{definition}
-%
-%\begin{question}
-% How do we go from $\langle e_1 \otimes f_1, e_2 \otimes f_2\rangle _{E\otimes _B F}$ to $
-% \langle f_1,\langle e_1, e_2\rangle _E f_2\rangle _F$ \label{q: tensorproduct}\\
-% This statement is still in the definition.
-%\end{question}
-%
-%%\begin{question}
-%%What is the meaning of `associative up to isomorphism'? Isomorphism of $F \circ E$ or of $A, B$ or $D$?
-%%\end{question}
-%
-%% \begin{MyExercise}
-%% \textbf{
-%% Show that the association $\phi \leadsto E_\phi$ (from the previous Example) is natural
-%% in the sense
-%% \begin{enumerate}
-%% \item $E_{\text{id}_A} \simeq A \in KK_f(A,A)$
-%% \item for $*$-algebra homomorphism $\phi: A \rightarrow B$ and $\psi: B \rightarrow C$ we have
-%% an isomorphism
-%% \begin{align}
-%% E_{\psi} \circ E_{\phi}\ \equiv\ E_{\phi} \otimes _B E_{\psi}\ \simeq\
-%% E_{\psi \circ \phi} \in KK_f(A,C)
-%% \end{align}
-%% \end{enumerate}
-%%}
-%% \begin{enumerate}
-%% \item $\text{id}_A: A \rightarrow A$.\\
-%% To construct $E_{\phi}\in KK_f(A,A)$, we let $E_{\phi}$ be $A$ with a natural right
-%% representation, so $\Rightarrow E_{\phi}\simeq A$.\\
-%% With an inner product, acting on $A$ from the left with $\phi$, $a', a\in A$\\
-%% $a'a = (\phi(a') a) \in A $, which is satisfied by $\text{id}_A$, so $\phi = \text{id}_A$.
-%% \item $a \cdot b \cdot c = \psi(\phi (a) \cdot b) \cdot c$ for $a \in A$, $b\in B$, and $c\in C$
-%% which is $\psi \circ \phi$
-%% \end{enumerate}
-%%\end{MyExercise}
-%
-%%\begin{MyExercise}
-%% \textbf{
-%% In the definition of Morita equivalence:
-%% \begin{enumerate}
-%% \item Check that $E \otimes _B F$ is a $A-D$ bimodule
-%% \item Check that $\langle \cdot,\cdot\rangle _{E\oplus _B F}$ defines a $D$ valued inner product
-%% \item Check that $\langle a^*(e_1 \otimes f_1), e_2 \otimes f_2\rangle _{E \otimes _B F} = \langle e_1 \otimes f_1, a(e_2 \otimes f_2)\rangle _{E \otimes _B F}$.
-%% \end{enumerate}
-%%}
-%% \begin{enumerate}
-%% \item $E \otimes _B F = E \otimes F / \{\sum_i e_i b_i \otimes f_i - e_i \otimes b_i f_i;
-%% e_i \in E_i, b_i \in B, f_i \in F\}$ the last part takes out all tensor product elements of
-%% $E$ and $F$ that don't preserver the left/right representation and that are duplicates.
-%% \item $\langle e_1, e_2\rangle _E \in B$ and $\langle f_1, f_2\rangle _F \in C$ by definition. So let $\langle e_1, e_2\rangle _E =b$. \\
-%% Then $\langle e_1 \otimes f_1, e_2 \otimes f_2\rangle _{E\otimes _B F} = \langle f_1, \langle e_1, e_2\rangle _E f_2\rangle _F =
-%% \langle f_1, b f_2\rangle _F \in C$
-%% \item Check Question \ref{q: tensorproduct}.\\
-%% But let $G := E\otimes _B F \in KK_f(A,C)$ then $\forall g_1, g_2 \in G$ and $a \in A$ we need
-%% by definition $\langle g_1, ag_2\rangle _G = \langle a^*g_1, g_2\rangle _G$ and we set $g_1 = e_1 \otimes f_1$ and
-%% $g_2 = e_2 \otimes f_2$ for some $e_1, e_2 \in E$ and $f_1, f_2 \in F$, or else
-%% $G \notin KK_f(A,C)$ which would violate the Kasparov product
-%% \end{enumerate}
-%% \end{MyExercise}
-%
-%\begin{definition}
-% Let $A$, $B$ be \textit{matrix algebras}. They are called \textit{Morita equivalent} if there
-% exists an $E \in KK_f(A, B)$ and an $F \in KK_f(B, A)$ such that:
-% \begin{align}
-% E \otimes _B F \simeq A \;\;\; \text{and} \;\;\; F \otimes _A E \simeq B
-% \end{align}
-% Where $\simeq$ denotes the isomorphism between Hilbert bimodules, note that $A$ or $B$ is a bimodule by
-% itself.
-%\end{definition}
-%
-%\begin{question}
-% Why are $E$ and $F$ each others inverse in the Kasparov Product? \\
-% They are each others inverse with respect to the Kasparov Product because we land in the same space as we started.
-% In the definition we have $E \in KK_f(A, B)$ we start from $A$ and $E \otimes _B F$ lands in $A$.\\
-% On the other hand we have $F \in KK_f(B, D)$ we start from $B$ and $F \otimes _A E$ lands in $B$.
-%\end{question}
-%
-%\begin{example}
-% \
-% \begin{itemize}
-% \item Hilber bimodule of $(A,A)$ is $A$
-% \item Let $E \in KK_f(A,B)$, we take $E \circ A = A\oplus _A E \simeq E$
-% \item we conclude, that $_A A_A$ is the identity in the Kasparov product (up to isomorphism)
-% \end{itemize}
-%\end{example}
-%
-%\begin{example}
-% Let $E = \mathbb{C}^n$, which is a $(M_n(\mathbb{C}), \mathbb{C})$ Hilbert bimodule with the
-% standard $\mathbb{C}$ inner product.\\
-% On the other hand let $F = \mathbb{C}^n$, which is a $(\mathbb{C}, M_n(\mathbb{C}))$ Hilbert
-% bimodule by right matrix multiplication with $M_n(\mathbb{C})$ valued inner product:
-% \begin{align}
-% \langle v_1, v_2\rangle =\bar{v_1}v_2^t \;\; \in M_n(\mathbb{C})
-% \end{align}
-% Now we take the Kasparov product of $E$ and $F$:
-% \begin{itemize}
-% \item $F\circ E\ =\ E\otimes _{\mathbb{C}}F\ \;\;\;\;\;\; \simeq \ M_n(\mathbb{C})$
-% \item $E\circ F\ =\ F\otimes _{M_n(\mathbb{C})}E\ \simeq\ \mathbb{C}$
-% \end{itemize}
-% $M_n(\mathbb{C})$ and $\mathbb{C}$ are Morita equivalent
-%\end{example}
-%
-%\begin{theorem}
-% Two matrix algebras are Morita Equivalent iff their their Structure spaces
-% are isomorphic as discreet spaces (have the same cardinality / same number of elements)
-%\end{theorem}
-%\begin{proof}
-% Let $A$, $B$ be \textit{Morita equivalent}. So there exists $_A E_B$ and $_B F_A$ with
-% \begin{align}
-% E \otimes _B F \simeq A \;\;\; \text{and} \;\;\; F \otimes _A E \simeq B
-% \end{align}
-% Consider $[(\pi _B, H)] \in \hat{B}$ than we construct a representation of $A$,
-% \begin{align}
-% \pi _A \rightarrow L(E \otimes _B H)\;\;\; \text{with} \;\;\; \pi _A(a) (e \otimes v) = a e \otimes w
-% \end{align}
-% \begin{question}
-% Is $E \simeq H$ and $F \simeq W$? \\
-% Not in particular, there is a theorem that all infinite dimensional Hilbert spaces are isomorphic.
-% Here we are looking at finite dimensional Hilbert spaces.\\
-% Another thing to is that $[\pi _B, H] \in \hat{B}$ and looking at Exercise \ref{ex: bimodule}
-% we know that $H$ is a bimodule of $B$, hence $E \otimes _B H\simeq A$, and for $[\pi _A, W]$
-% the same.
-% \end{question}
-% \textit{vice versa}, consider $[(\pi _A, W)] \in \hat{A}$ we can construct $\pi _B$
-% \begin{align}
-% \pi _B: B \rightarrow L(F \otimes _A W) \;\;\; \text{and}\;\;\; \pi _B(b) (f\otimes w) = bf\otimes w
-% \end{align}
-% These maps are each others inverses, thus $\hat{A} \simeq \hat{B}$
-%\end{proof}
-%
-%%\begin{MyExercise}
-%% \textbf{
-%% Fill in the gaps in the above proof:
-%% \begin{enumerate}
-%% \item show that the representation of $\pi _A$ defined is irreducible iff $\pi _B$ is.
-%% \item Show that the association of the class $[\pi _A]$ to $[\pi _B]$ is independent
-%% of the choice of representatives $\pi _A$ and $\pi _B$
-%% \end{enumerate}
-%%}
-%%
-%% \begin{enumerate}
-%% \item $(\pi _B, H)$ is irreducible means $H \neq \emptyset$ and only $\emptyset$ or $H$
-%% is invariant under the Action of $B$ on $H$.
-%% Than $E\otimes _B H$ cannot be empty, because also $E$ preserves left representation of $A$
-%% and also $E\otimes _B H \simeq A$.
-%% \item The important thing is that $[\pi _A] \in \hat{A}$ respectively $[\pi _B] \in \hat{B}$,
-%% hence any choice of representation is irreducible, because the structure space denotes all unitary
-%% equivalence classes of irreducible representations.
-%% \end{enumerate}
-%%\end{MyExercise}
-%
-% \begin{lemma}
-% The matrix algebra $M_n(\mathbb{C})$ has a unique irreducible representation (up to isomorphism)
-% given by the defining representation on $\mathbb{C}^n$.
-%\end{lemma}
-%\begin{proof}
-% We know $\mathbb{C}^n$ is a irreducible representation of $A= M_n(\mathbb{C})$. Let $H$ be irreducible
-% and of dimension $k$, then we define a map
-% \begin{align}
-% \phi : A\oplus...\oplus A &\rightarrow H^* \\
-% (a_1,...,a_k) &\mapsto e^1\circ a_1^t+...+e^k\circ a_k^t
-% \end{align}
-% With $\{e^1,...,e^k\}$ being the basis of the dual space $H^*$ and $(\circ)$ being the pre-composition
-% of elements in $H^*$ and $A$ acting on $H$. This forms a morphism of $M_n(\mathbb{C})$ modules,
-% provided a matrix $a \in A$ acts on $H^*$ with $v\mapsto v\circ a^t$ ($v\in H^*$).
-% Furthermore this morphism is surjective, thus making the pullback $\phi ^*:H\mapsto (A^k)^*$ injective.
-% Now identify $(A^k)^*$ with $A^k$ as a $A$-module and note that
-% $A=M_n(\mathbb{C}) \simeq \oplus ^n \mathbb{C}^n$ as a n A module.
-% It follows that $H$ is a submodule of $A^k \simeq \oplus ^{nk}\mathbb{C}$. By irreducibility
-% $H \simeq \mathbb{C}$.
-%\end{proof}
-%
-%\begin{example}
-% Consider two matrix algebras $A$, and $B$.
-% \begin{align}
-% A = \bigoplus ^N_{i=1} M_{n_i}(\mathbb{C}) \;\;\; B = \bigoplus ^M_{j=1} M_{m_j}(\mathbb{C})
-% \end{align}
-% Let $\hat{A} \simeq \hat{B}$ that implies $N=M$ and define $E$ with $A$ acting by block-diagonal
-% matrices on the first tensor and B acting in the same way on the second tensor. Define $F$ vice versa.
-% \begin{align}
-% E:= \bigoplus _{i=1}^N \mathbb{C}^{n_i} \otimes \mathbb{C}^{m_i} \;\;\;
-% F:= \bigoplus _{i=1}^N \mathbb{C}^{m_i} \otimes \mathbb{C}^{n_i}
-% \end{align}
-% Then we calculate the Kasparov product.
-% \begin{align}
-% E \otimes _B F &\simeq \bigoplus _{i=1}^N (\mathbb{C}^{n_i}\otimes\mathbb{C}^{m_i})
-% \otimes _{M_{m_i}(\mathbb{C})} (\mathbb{C}^{m_i}\otimes\mathbb{C}^{n_i}) \\
-% &\simeq \bigoplus _{i=1}^N \mathbb{C}^{n_i}\otimes
-% \left(\mathbb{C}^{m_i}\otimes _{M_{m_i}(\mathbb{C})}\mathbb{C}^{m_i}\right)
-% \oplus \mathbb{C}^{n_i} \\
-% &\simeq \bigoplus _{i=1}^N \mathbb{C}^{m_i}\otimes\mathbb{C}^{n_i} \simeq A
-% \end{align}
-% and from $F \otimes _A E \simeq B$.
-%\end{example}
-%
-%We conclude that.
-%\begin{itemize}
-% \item There is a duality between finite spaces and Morita equivalence classes of matrix algebras.
-% \item By replacing $*$-homomorphism $A\rightarrow B$ with Hilbert bimodules $(A,B)$ we introduce
-% a richer structure of morphism between matrix algebras.
-%\end{itemize}
-%
-%\subsection{Noncommutative Geometric Spaces }
-%\subsubsection{Exercises}
-%
-%%\begin{MyExercise}
-%%\textbf{
-%% Make the proof of the last theorem (see week4.pdf) explicit for $N=3$.
-%%}\newline
-%%
-%% For the C* algebra we have $A=\mathbb{C}^3$
-%% For $H$ we have $H = (\mathbb{C}^2)^{\oplus 3} = H_2 \oplus H_2^1 \oplus H_2^2$.
-%% The symmetric operator $D$ acting on $H$ and the representation $\pi (a)$:
-%% \begin{align}
-%% \pi((a(1), a(2), a(3)) &=
-%% \begin{pmatrix}
-%% a(1) & 0 \\ 0 & a(2)
-%% \end{pmatrix} \oplus
-%% \begin{pmatrix}
-%% a(1) & 0 \\ 0 & a(3)
-%% \end{pmatrix} \oplus
-%% \begin{pmatrix}
-%% a(2) & 0 \\ 0 & a(2)
-%% \end{pmatrix} \nonumber \\
-%% & =
-%% \begin{pmatrix}
-%% a(1) & 0 & 0 & 0 & 0 & 0 \\
-%% 0 & a(2) & 0 & 0 & 0 & 0 \\
-%% 0 & 0 & a(1) & 0 & 0 & 0 \\
-%% 0 & 0 & 0 & a(3) & 0 & 0 \\
-%% 0 & 0 & 0 & 0 & a(2) & 0 \\
-%% 0 & 0 & 0 & 0 & 0 & a(3)
-%% \end{pmatrix} \\
-%% D &=
-%% \begin{pmatrix}
-%% 0 & x_1 \\ x_1 & 0
-%% \end{pmatrix} \oplus
-%% \begin{pmatrix}
-%% 0 & x_2 \\ x_2 & 0
-%% \end{pmatrix} \oplus
-%% \begin{pmatrix}
-%% 0 & x_3 \\ x_3 & 0
-%% \end{pmatrix} \nonumber \\
-%% &=
-%% \begin{pmatrix}
-%% 0 & x_1 & 0 & 0 & 0 & 0 \\
-%% x_1 & 0 & 0 & 0 & 0 & 0 \\
-%% 0 & 0 & 0 & x_2 & 0 & 0 \\
-%% 0 & 0 & x_2 & 0 & 0 & 0 \\
-%% 0 & 0 & 0 & 0 & 0 & x_3 \\
-%% 0 & 0 & 0 & 0 & x_3 & 0 \\
-%% \end{pmatrix} \\
-%% \end{align}
-%% Then the norm of the commutator would be the largest eigenvalue
-%% \begin{align}
-%% &||[D, \pi(a)]|| = ||D\pi(a) - \pi(a)D||\nonumber\\
-%% &=
-%% \left|\left|
-%% \setlength{\arraycolsep}{0.1cm}
-%% \renewcommand{\arraystretch}{0.1}
-%% \begin{pmatrix}
-%% 0 & x_1(a(2)-a(1)) & 0 & 0 & 0 & 0 \\
-%% -x_1(a(2)-a(1)) & 0 & 0 & 0 & 0 & 0 \\
-%% 0 & 0 & 0 & x_2(a(3)-a(1)) & 0 & 0 \\
-%% 0 & 0 & -x_2(a(3)-a(1)) & 0 & 0 & 0 \\
-%% 0 & 0 & 0 & 0 & 0 & x_3(a(3)-a(2)) \\
-%% 0 & 0 & 0 & 0 & -x_3(a(2)-a(3)) & 0 \\
-%% \end{pmatrix}\right|\right| \label{skew matrix}
-%% \end{align}
-%%The matrix in Equation \ref{shew matrix} is a skew symmetric matrix its eigenvalues
-%%are $i\lambda_1, i\lambda_2, i\lambda_3, i\lambda_4$, where the $\lambda$'s are on the
-%%upper and lower diagonal check \url{https://en.wikipedia.org/wiki/Skew-symmetric_
-%%matrix#Skew-symmetrizable_matrix}. The matrix norm of would be the maximum of the norm of
-%%the larges eigenvalues:
-%%\begin{align}
-%% ||[D, \pi(a)]|| = \max_{a\in A}\{&x_1|a(2)-a(1)|,\\
-%% &x_2|(a(3)-a(1))|,\nonumber\\
-%% &x_3|(a(3)-a(2))|,\}\nonumber
-%%\end{align}
-%%The metric is then:
-%%\begin{align}
-%% d =
-%% \begin{pmatrix}
-%% 0 & a(1)-a(2) & a(1)-a(3)\\
-%% a(2)-a(1) & 0 & a(2)-a(3)\\
-%% a(3)-a(1) & a(3)-a(2) & 0
-%% \end{pmatrix}
-%%\end{align}
-%%\end{MyExercise}
-%
-%%\begin{MyExercise}
-%% \textbf{
-%% Compute the metric on the space of three points given by $d_{ij} =
-%% \sup_{a\in A}\{|a(i) - a(j)|: ||[D, \pi(a)]|| \leq 1\}$ for the set of data
-%% $A = \mathbb{C}^3$ acting in the defining representation $H = \mathbb{C}^3$, and
-%% \begin{align}
-%% D =
-%% \begin{pmatrix}
-%% 0 & d^{-1} & 0 \\
-%% d^{-1} & 0 & 0 \\
-%% 0 & 0 & 0
-%% \end{pmatrix}
-%% \end{align}
-%% for some $d \in \mathbb{R}$
-%%}\newline
-%%
-%% We have $A=\mathbb{C}^3$, $H=\mathbb{C}^3$ and $D$ from above, then
-%%
-%% \begin{align}
-%% ||[D, \pi(a)]|| &= d^{-1}\left|\left|
-%% \begin{pmatrix}
-%% 0 & a(2)-a(1) & 0 \\
-%% -(a(2)-a(1)) & 0 & 0 \\
-%% 0 & 0 & 0
-%% \end{pmatrix} \right|\right|
-%% \end{align}
-%% The metric is then
-%% \begin{align}
-%% d =
-%% \begin{pmatrix}
-%% 0 & a(1)-a(2) & a(1) \\
-%% a(2)-a(1) & 0 & a(2) \\
-%% -a(1) & -a(2) & 0
-%% \end{pmatrix}
-%% \end{align}
-%%\end{MyExercise}
-%
-%%\begin{MyExercise}
-%% \textbf{
-%% Show that $d_{ij}$ from Equation \ref{ext metric} is a metric on $\hat{A}$ by
-%% establishing that:
-%% \begin{align}
-%% d_{ij} &= 0\;\;\; \Leftrightarrow \;\;\; i=j \label{metric 1} \\
-%% d_{ij} &= d_{ji} \label{metric 2}\\
-%% d_{ij} &\leq d_{ik} + d_{kj} \label{metric 3}
-%% \end{align}
-%% \begin{equation} \label{ext metric}
-%% d_{ij} = \sup_{a\in A}\big\{|\text{Tr}(a(i)) - \text{Tr}((a(j))|: ||[D, a]|| \leq 1\big\}
-%% \end{equation}
-%%}\newline
-%%
-%%For Equation \ref{metric 1} set $i=j$ in \ref{ext metric}.
-%%\begin{align}
-%% d_{ii} &= \sup_{a \in A}\{|\text{Tr}(a(i)) - \text{Tr}((a(i))|: ||[D, a]|| \leq
-%% 1\big\} \\
-%% &= \sup_{a \in A}\{0: ||[D, a]|| \leq 1\big\} = 0
-%%\end{align}
-%%For Equation \ref{metric 2} obviously we have the commuting property of
-%%addition.
-%%\newline
-%%For Equation \ref{metric 3}, for $k=j$ then $d_{kj} = 0$ and the equality
-%%holds. For $i = k$ then $d_{ik} = 0$ and equality holds. Else set $d_{ik} =
-%%1$ and $d_{kj} = 1$ then $d_{ij} = 1 \leq d_{ik} + d_{kj} = 2$
-%%\end{MyExercise}
-%
-%\subsubsection{Properties of Matrix Algebras}
-%\begin{lemma}
-% If $A$ is a unital C* algebra that acts faithfully on a finite
-% dimensional Hilbert space, then $A$ is a matrix algebra of the Form:
-% \begin{equation}
-% A \simeq \bigoplus _{i=1}^N M_{n_i}(\mathbb{C})
-% \end{equation}
-%\end{lemma}
-%\begin{proof}
-% Since $A$ acts faithfully on a Hilbert space, then $A$ is a C*
-% subalgebra of a matrix algebra $L(H) = M_{\dim (H)}(\mathbb{C}
-% \Rightarrow A \simeq \text{Matrix algebra}$.
-%\end{proof}
-%
-%\begin{question}
-% What does the author mean when he sais 'acts faithfully on a
-% Hilbertspace`? Then the representation is fully reducible, or that the
-% presentation is irreducible?
-% \newline
-%
-% For a *-representation 'faithful` if it is injective. For a
-% *-homomorphism 'faithful` means one-to-one correspondance
-%\end{question}
-%
-%\begin{example}
-% $A = M_n(\mathbb{C})$ and $H=\mathbb{C}^n$, $A$ acts on $H$ with matrix
-% multiplication and standard inner product. $D$ on $H$ is a hermitian
-% matrix $n\times n$ matrix.
-%\end{example}
-%
-%$D$ is referred to as a finite Dirac operator as in as its $\infty$
-%dimensional on Riemannian Spin manifolds coming in Chapter 4.
-%\newline
-%
-%Now can introduce a 'differential 'geometric structure` on the finite space X
-%with the \textbf{devided difference}
-%\begin{equation}
-% \frac{a(i)-a(j)}{d_{ij}}
-%\end{equation}
-%for each pair $i$, $j$ $\in X$ the finite dimensional discrete space $X$.
-%This appears in the entries in the commutator $[D, a]$ in the above
-%exercises.
-%
-%\begin{definition}
-% Given an finite spectral triple $(A, H, D)$, the $A$-bimodule of
-% Connes' differential one-forms is:
-% \begin{equation}
-% \Omega _D ^1 (A) := \left\{ \sum _k a_k[D, b_k]: a_k, b_k \in A \right\}
-% \end{equation}
-%\end{definition}
-%
-%\begin{question}
-% Is the Conne's differential one form the set of all '1st order
-% differential operators` given $A$, that act on $H$?
-%\end{question}
-%Then there is a map $d:A\rightarrow \Omega _D ^1 (A)$, $d = [D, \cdot]$.
-%%\begin{MyExercise}
-%% \textbf{
-%% Verify that 'd` is a derivation of the C* algebra
-%% \begin{align}
-%% d(ab) = d(a)b + ad(b) \\
-%% d(a^*) = -d(a)^*
-%% \end{align}
-%%}\newline
-%%
-%% For the record $d(\cdot) = [D, \cdot]$, then we have
-%% \begin{enumerate}
-%% \item
-%% \begin{align}
-%% d(ab) &= [D, ab] = [D, a]b + a[D,b]\\
-%% &= d(a)b + ad(b)
-%% \end{align}
-%% \item
-%% \begin{align}
-%% d(a^*) &= [D, a^*] = Da^* - a^*D \\
-%% &=-(D^*a - aD^*) = -[D^*, a] \\
-%% &= -d(a)^*
-%% \end{align}
-%% \end{enumerate}
-%%\end{MyExercise}
-%%\begin{MyExercise}
-%% \textbf{
-%% Verify that $\Omega _D^1 (A)$ is an $A$-bimodule by rewriting
-%% }
-%% \begin{align}
-%% a(a_k[D, b_k])b = \sum_k a'_k[D, b'_k] \;\;\;\; a'_k, b'_k \in A
-%% \end{align}
-%% \newline
-%%
-%% Begin
-%% \begin{align}
-%% a(a_k[D, b_k])b &= aa_k(Db_k - b_k D) b = \\
-%% &= aa_k(Db_k b - b_k D b) = aa_k(Db_k b - b_k Db - b_kbD +b_kbD)=\\
-%% &= aa_k(Db_kb-b_kbD + b_k b D - b_k D b) = \\
-%% &= aa_k [D, b_kb] + aa_k b [D, b]=\\
-%% &= \sum _k a_k' [D, b_k']
-%% \end{align}
-%%
-%%\end{MyExercise}
-%
-%\begin{lemma}
-% Let $(A, H, D) = (M_n(\mathbb{C}, \mathbb{C}^n, D)$, with $D$ a hermitian
-% $n\times n$ matrix. If $D$ is not a multiple of the identity then:
-% \begin{equation}
-% \Omega _D ^1 (A) \simeq M_n(\mathbb{C}) = A
-% \end{equation}
-%\end{lemma}
-%
-%\begin{proof}
-% Assume $D = \sum _i \lambda _i e_{ii}$ (diagonal), $\lambda _i \in \mathbb{R}$ and
-% $\{e_{ij}\}$ the basis of $M_n(\mathbb{C}$. For fixed $i$, $j$ choose $k$
-% such that $\lambda _k \neq \lambda _j$ then
-% \begin{align} \label{basis}
-% \left(\frac{1}{\lambda _k - \lambda _j} e_{ik}\right) [D, e_{kj}] =
-% e_{ij}
-% \end{align}
-% $e_{ij}\in \Omega _D ^1 (A)$ by the above definition. And $\Omega _D ^1
-% (A) \subset L(\mathbb{C}^n) = H \simeq M_n(\mathbb{C}) = A$
-%\end{proof}
-%
-%%\begin{MyExercise}
-%% \textbf{
-%% Consider $(A=\mathbb{C}^2, H=\mathbb{C}^2,
-%% D = \begin{pmatrix} 0 & \lambda \\ \bar{\lambda} & 0
-%% \end{pmatrix})$ with $\lambda \neq 0$. Show that $\Omega _D^1(A)
-%% \simeq M_2(\mathbb{C})$
-%% }
-%%\newline
-%%
-%% Because of the Hilbert Basis $D$ can be extended in terms of
-%% the basis of $M_2(\mathbb{C})$, plugging this into Equation
-%% \ref{basis} will get us the same cyclic result, thus
-%% $\Omega _D^1(A) \simeq M_2(\mathbb{C})$
-%%\
-%%\end{MyExercise}
-%
-%\subsubsection{Morphisms Between Finite Spectral Triples}
-%\begin{definition}
-% two finite spectral tripes $(A_1, H_1, D_1)$ and $(A_2, H_2, D_2)$ are
-% called unitarily equivalent if
-% \begin{itemize}
-% \item $A_1 = A_2$
-% \item $\exists \;\; U: H_1 \rightarrow H_2$, unitary with
-% \begin{enumerate}
-% \item $U\pi_1(a)U^* = \pi_2(a)$ with $a \in A_1$
-% \item $UD_1 U^* = D_2$
-% \end{enumerate}
-% \end{itemize}
-%\end{definition}
-%
-%Some remarks
-%\begin{itemize}
-% \item the above is an equivalence relation
-% \item spectral unitary equivalence is given by the unitaries of the
-% matrix algebra itself
-% \item for any such $U$ then $(A, H, D) \sim (A, H, UDU^*)$
-% \item $UDU^* = D + U[D, U^*]$ of the form of elements in
-% $\Omega _D^1 (A)$.
-%\end{itemize}
-%
-%%\begin{MyExercise}
-%% \textbf{
-%% Show that the unitary equivalence between finite spectral
-%% triples is a equivalence relation
-%%}\newline
-%%
-%% An equivalence relation needs to satisfy reflexivity, symmetry
-%% transitivity.
-%% Let $(A_1, H_1, D_1)$, $(A_2, H_2, D_2)$ and $(A_3, H_3, D_3)$
-%% be three finite spectral triples.
-%% \newline
-%%
-%% For reflexivity $(A_1, H_1, D_1) \sim (A_1, H_1, D_1)$. So there
-%% exists a $U: H_1 \rightarrow H_1$ unitary, which is the identity
-%% and always exists.
-%% \newline
-%%
-%% For symmetry we need
-%% \begin{align}
-%% (A_1, H_1, D_1) \sim (A_2, H_2, D_2) \Leftrightarrow
-%% (A_2, H_2, D_2) \sim (A_1, H_1, D_1)
-%% \end{align}
-%% because $U$ is unitary:
-%% \begin{align}
-%% &U\pi_1(a)U^* = \pi_2(a) \;\;\; | \cdot U^*\boxdot U \\
-%% &U^*U\pi_1(a)U^*U = \pi_1(a) = U^*\pi_2(a)U \\
-%% \end{align}
-%% The same with the symmetric operator $D$.
-%% \newline
-%%
-%% For transitivity we need
-%% \begin{align}
-%% (A_1, H_1, D_1) &\sim (A_2, H_2, D_2) \;\;\; \text{and} \;\;\;
-%% (A_2, H_2, D_2) \sim (A_3, H_3, D_3) \\
-%% &\Rightarrow (A_1, H_1, D_1) \sim (A_3, H_3, D_3)
-%% \end{align}
-%% There are two unitary maps $U_{12}:H_1 \rightarrow H_2$ and
-%% $U_{23}: H_2 \rightarrow H_3$ then
-%% \begin{align}
-%% U_{23}U_{12} \pi_1(a) U^*_{12}U^*_{23} &= U_{23}
-%% \pi_2(a) U_23^* \\
-%% &= \pi_3(a) \\
-%% U_{23}U_{12} D_1U^*_{12}U^*_{23} &= U_{23}
-%% D_2 U_23^* \\
-%% &= D_3
-%% \end{align}
-%%\end{MyExercise}
-%
-%Extending the this relation we look again at the notion of equivalence from
-%Morita equivalence of Matrix Algebras.
-%\newline
-%
-%\begin{definition}
-% Let $A$ be an algebra. We say that $I \subset A$, as a vector space, is a
-% right(left) ideal if $ab \in I$ for $a \in A$ and $b\in I$ (or $ba \in
-% I$, $b\in I$, $a\in A$). We call a left-right ideal simply an ideal.
-%\end{definition}
-%
-%Given a Hilbert bimodule $E \in KK_f(B, A)$ and $(A, H, D)$ we construct
-%a finite spectral triple on $B$, $(B, H', D')$
-%\begin{equation}
-% H' = E \otimes _A H
-%\end{equation}
-%We might define $D'$ with $D'(e \otimes \xi) = e\otimes D\xi$, thought this
-%would not satisfy the ideal defining the balanced tensor product over $A$,
-%which is generated by elements of the form
-%\begin{align}
-% e a \otimes \xi - e\otimes a \xi ;\;\;\;\; e\in E, a\in A, \xi \in H
-%\end{align}
-%This inherits the left action on $B$ from $E$ and has a $\mathbb{C}$
-%valued inner product space. $B$ also satisfies the ideal.
-%\begin{equation}
-% D'(e\otimes \xi) = e \otimes D \xi + \nabla (e) \xi \;\;\;\; e\in
-% E, a\in A
-%\end{equation}
-%Where $\nabla$ is called the \textit{connection on the right A-module E}
-%associated with the derivation $d=[D, \cdot]$ and satisfying the
-%\textit{Leibnitz Rule} which is
-%\begin{equation}
-% \nabla(ae) = \nabla(e)a + e \otimes [D, a] \;\;\;\;\; e\in E,\; a\in A
-%\end{equation}
-%Then $D'$ is well defined on $E \otimes _A H$:
-%\begin{align}
-% D'(ea \otimes \xi - e \otimes a \xi) &= D'(ea \otimes \xi) - D'(e
-% \otimes \xi) \\
-% &= ea\otimes D\xi + \nabla(ae) \xi - e \otimes D(a\xi ) - \nabla (e)a
-% \xi \\
-% &= 0.
-%\end{align}
-%With the information thus far we can prove the following theorem
-%\begin{theorem}
-% If $(A, H, D)$ a finite spectral triple, $E \in KK_f(B, A)$.
-% Then $(V, E\otimes _A H, D')$ is a finite spectral triple, provided that
-% $\nabla$ satisfies the compatibility condition
-% \begin{equation}
-% \langle e_1, \nabla e_2 \rangle _E - \langle \nabla e_1, e_2
-% \rangle _E = d\langle e_1, e_2 \rangle _E \;\;\;\; e_1, e_2 \in E
-% \end{equation}
-%\end{theorem}
-%\begin{proof}
-% $E\otimes _A H$ was shown in the previous subsection (text before the
-% theorem). The only thing left is to show that $D'$ is a symmetric
-% operator, this we can just compute. Let $e_1, e_2 \in E$ and $\xi _1,
-% \xi _2 \in H$ then
-% \begin{align}
-% \langle e_1 \otimes \xi _1, D'(e_2 \otimes \xi_2)\rangle _{E\otimes _A H} &=
-% \langle \xi _1, \langle e_1, \nabla e_2\rangle _E \xi _2\rangle + \langle \xi _1 , \langle e_1, e_2\rangle _E D\xi
-% _2\rangle _H \\
-% &= \langle \xi _1, \langle \nabla e_1, e_2\rangle _E \xi _2\rangle _H + \langle \xi _1, d\langle e_1, e_2\rangle _E
-% \xi _2\rangle _H \\
-% &+ \langle D\xi _1,\langle e_1, e_2\rangle _E \xi _2\rangle _H - \langle \xi _1, [D, \langle e_1, e_2\rangle _E] \xi
-% _2 \rangle _H \\
-% &= \langle D'(e_1 \otimes \xi _1), e_2 \otimes \xi _2\rangle _{E \otimes _A H}
-% \end{align}
-%\end{proof}
-%
-%%\begin{MyExercise}
-%% \textbf{
-%% Let $\nabla$ and $\nabla'$ be two connections on a right $A$-module
-%% $E$. Show that $\nabla - \nabla'$ is a right $A$-linear map
-%% $E \rightarrow E\otimes _A \Omega _D^1(A)$
-%%}\newline
-%%
-%% Both $\nabla$ and $\nabla'$ need to satisfy the Leiblitz rule, so
-%% let's see if $\nabla - \nabla'$ does.
-%%
-%% \begin{align}
-%% \nabla(ea)-\nabla'(ea)&=\nabla(e) + e\otimes[D, a]\\
-%% &-(\nabla'(e)a + e\otimes[D',a])\\
-%% &=\bar{\nabla}a + e\otimes(Da-aD-D'a+aD')\\
-%% &=\bar{\nabla}a + e\otimes((D-D')a-a(D-D'))\\
-%% &=\bar{\nabla}a + e\otimes[D', a]\\
-%% &=\bar{\nabla}(ea)
-%% \end{align}
-%% Therefore $\nabla-\nabla'$ is a linear map.
-%%\end{MyExercise}
-%
-%%\begin{MyExercise}
-%% \textbf{
-%% Construct a finite spectral triple $(A, H', D')$ from $(A, H, D)$
-%% \begin{enumerate}
-%% \item show that the derivation $d(\cdot):A \rightarrow A\otimes _A
-%% \Omega_D^1(A)=\Omega_D^1(A)$ is a connection on $A$
-%% considered a right $A$-module
-%% \item Upon identifying $A\otimes_A H\simeq H$, what is $D'$
-%% when the connection is $d(\cdot)$.
-%% \item Use 1) and 2) to show that any connection $\nabla:
-%% A\rightarrow A\otimes_A \Omega_D^1(A)$ is given by
-%% \begin{align}
-%% \nabla = d + \omega
-%% \end{align}
-%% where $\omega \in \Omega_D^1(A)$
-%% \item Upon identifying $A\otimes_A H \simeq H$, what is the
-%% difference operator $D'$ with the connection on $A$ given by
-%% $\nabla = d + \omega$
-%% \end{enumerate}
-%%}
-%% \begin{enumerate}
-%% \item $\nabla(e \cdot a) = d(a)$
-%% \item
-%% $D'(a\xi) = a(D\xi) + (\nabla a) \xi = D(a\xi)$
-%% \item Use the identity element $e \in A$\\
-%% $\nabla (e\cdot a) = \nabla(e) a + 1 \otimes d(a)=d(a)
-%% \nabla(e) a$
-%% \item $D'(a\otimes \xi) = D'(a \xi) = a(D\xi) + (\nabla a)\xi =
-%% a(D\xi) + \nabla(e \cdot a) \xi \\
-%% = D(a\xi) + \nabla(e) (a\xi)$
-%% \end{enumerate}
-%%\end{MyExercise}
-%
-%\subsubsection{Graphing Finite Spectral Triples}
-%\begin{definition}
-% A \textit{graph} is a ordered pair $(\Gamma ^{(0)}, \Gamma ^{(1)})$.
-% Where $\Gamma ^{(0)}$ is the set of vertices (nodes) and $\Gamma ^{(1)}$
-% a set of pairs of vertices (edges)
-%\end{definition}
-%\begin{figure}[h!]
-% \centering
-%\begin{tikzpicture}[
-% mass/.style = {draw,circle, minimum size=0.2cm, inner sep=0pt, thick},
-% spring/.style = {decorate,decoration={zigzag, pre length=1cm,post length=1cm,segment length=5pt}},]
-% \node[mass] (m1) at (1,1.5) {};
-% \node[mass] (m2) at (-1,1.5) {};
-% \node[mass] (m3) at (0,0) {};
-%
-% \draw (m1) -- (m2);
-% \draw (m1) -- (m3);
-% \draw (m2) -- (m3);
-% \end{tikzpicture}
-% \caption{A simple graph with three vertices and three edges}
-%\end{figure}
-%%\begin{MyExercise}
-%% \textbf{
-%% Show that any finite-dimensional faithful representation $H$ of a matrix
-%% algebra $A$ is completely reducible. To do that show that the complement
-%% $W^{\perp}$ of an $A$-submodule $W\subset H$ is also an $A$-submodule
-%% of $H$.
-%%}\newline
-%%
-%% $A\simeq \bigoplus_{i=1}^N M_{n_i}(\mathbb{C})$ is the matrix algebra
-%% then $H$ is a Hilbert $A$-bimodule and $W$ a submodule of $A$.
-%% Because we have $H = W \cup W^{\perp}$, then $W^{\perp}$ is naturally a
-%% $A$-submodule, because elements in $W^{\perp}$ need to satisfy the
-%% bimodularity.
-%%\end{MyExercise}
-%\begin{definition}
-% A $\Lambda$-decorated graph is given by an ordered pair $(\Gamma,
-% \Lambda)$ of a finite graph $\Gamma$ and a set of positive integers
-% $\Lambda$ with the labeling
-% \begin{itemize}
-% \item of the vetices $v\in \Gamma ^{(0)}$ given by $n(\nu) \in
-% \Lambda$
-% \item of the edges $e = (\nu _1, \nu _2) \in \Gamma ^{(1)}$ by
-% operators
-% \begin{itemize}
-% \item $D_e: \mathbb{C}^{n(\nu _1)} \rightarrow
-% \mathbb{C}^{n(\nu _2)}$
-% \item and $D_e^*: \mathbb{C}^{n(\nu _2)} \rightarrow
-% \mathbb{C}^{n(\nu _1)}$ its conjugate traspose
-% (pullback?)
-% \end{itemize}
-% \end{itemize}
-% such that
-% \begin{equation}
-% n(\Gamma ^{(0)}) = \Lambda
-% \end{equation}
-%\end{definition}
-%\begin{question}
-% Would then $D_e$ be the pullback?
-%\end{question}
-%\begin{question}
-% These graphs are important in the next chapter I should look
-% into it more, I don't understand much here, specific
-% how to construct them with the abstraction of a spectral triple...
-%\end{question}
-%
-%The operator $D_e$ between $\textbf{n}_i$ and $\textbf{n}_j$ add up to
-%$D_{ij}$
-%\begin{align}
-% D_{ij} = \sum\limits_{\substack{e = (\nu _1, \nu _2) \\ n(\nu _1) =
-% \textbf{n}_i \\ n(\nu _2) = \textbf{n}_j}} D_e
-%\end{align}
-%
-%\begin{theorem}
-% There is a on to one correspondence between finite spectral triples
-% modulo unitary equivalence and $\Lambda$-decorated graphs, given by
-% associating a finite spectral triples $(A, H, D)$ to a $\Lambda$ decorated
-% graph $(\Gamma, \Lambda)$ in the following way:
-% \begin{equation}
-% A = \bigoplus _{n\in \Lambda} M_n(\mathbb{C}); \;\;\;
-% H = \bigoplus _{\nu \in \Gamma ^{(0)}} \mathbb{C}^{n(\nu)}; \;\;\;
-% D = \sum _{e \in \Gamma ^{(1)}} D_e + D_e^*
-% \end{equation}
-%\end{theorem}
-% \begin{figure}[h!]
-% \centering
-% \begin{tikzpicture}[
-% mass/.style = {draw,circle, minimum size=0.3cm, inner sep=0pt, thick},
-% ]
-%
-% \node[mass, label={\textbf{n}}] (m1) at (1,0) {};
-% \draw (m1) to [out=330, in=210, looseness=25] node[above] {$D_e$} (m1);
-% \end{tikzpicture}
-% \caption{A $\Lambda$-decorated Graph of $(M_n(\mathbb{C}), \mathbb{C}^n,
-% D = D_e + D_e^*)$}
-%\end{figure}
-%
-%%\begin{MyExercise}
-%% \textbf{
-%% Draw a $\Lambda$ decorated graph corresponding to the spectral triple
-%% $(A=\mathbb{C}^3, H=\mathbb{C}^3, D=\begin{pmatrix}0 & \lambda & 0\\
-%% \bar{\lambda} &0 &0 \\ 0&0&0\end{pmatrix})$
-%%}\newline
-%%
-%%\centering
-%%\begin{tikzpicture}[
-%% mass/.style = {draw,circle, minimum size=0.4cm, inner sep=0pt, thick},
-%% spring/.style = {decorate,decoration={zigzag, pre length=1cm,post length=1cm,segment length=5pt}},]
-%% \node[mass] (m1) at (-1,1.5) {\textbf{1}};
-%% \node[mass] (m2) at (1,1.5) {\textbf{2}};
-%% \node[mass] (m3) at (3,1.5) {\textbf{3}};
-%%
-%% \draw[style=thick, -] (1.1,1.7) -- (-1.1,1.7);
-%% \draw[style=thick, -] (1.1,1.3) -- (-1.1,1.3);
-%% \end{tikzpicture}
-%% % \captionof{figure}{Solution}
-%%\end{MyExercise}
-%%\begin{MyExercise}
-%% \textbf{
-%% Use $\Lambda$-decorated graphs to classify all finite spectral triples
-%% (modulo unitary equivalence) on the matrix algebra
-%% $A=\mathbb{C}\oplus M_2(\mathbb{C})$
-%%}\newline
-%%
-%% \centering
-%%\begin{tikzpicture}[
-%% mass/.style = {draw,circle, minimum size=0.4cm, inner sep=0pt, thick},
-%% spring/.style = {decorate,decoration={zigzag, pre length=1cm,post length=1cm,segment length=5pt}},]
-%% \node[mass] (m1) at (-1,1) {\textbf{1}};
-%% \node[mass] (m2) at (1,1) {\textbf{2}};
-%% \node[mass] (m3) at (3,1) {\textbf{3}};
-%%
-%% \node[mass] (m4) at (-1,0) {\textbf{1}};
-%% \node[mass] (m5) at (1,0) {\textbf{2}};
-%% \node[mass] (m6) at (3,0) {\textbf{3}};
-%%
-%% \node[mass] (m7) at (-1,-1) {\textbf{1}};
-%% \node[mass] (m8) at (1,-1) {\textbf{2}};
-%% \node[mass] (m9) at (3,-1) {\textbf{3}};
-%%
-%% \node[mass] (m10) at (-1,-2) {\textbf{1}};
-%% \node[mass] (m11) at (1,-2) {\textbf{2}};
-%% \node[mass] (m12) at (3,-2) {\textbf{3}};
-%%
-%% \draw[style=thick, -] (1.1,0.2) -- (-1.1,0.2);
-%% \draw[style=thick, -] (1.1,-0.2) -- (-1.1,-0.2);
-%% \draw[style=thick, -] (m7) to [out=330, in=210, looseness=10] node[above] {} (m7);
-%% \draw[style=thick, -] (m10) -- (m11) ;
-%%
-%%\end{tikzpicture}
-%%% \captionof{figure}{Solution $A=M_3(\mathbb{C})$}
-%%\end{MyExercise}
-%\subsubsection{Graph Construction of Finite Spectral Triples}
-%\textbf{Algebra:}We know if a acts on a finite dimensional Hilbert space then
-%this C* algebra is isomorphic to a matrix algebra so $A \simeq
-%\bigoplus_{i=1}^{N}M_{n_i}(\mathbb{C})$. Where $i\in
-%\hat{A}$ represents an equivalence class and runs from $1$ to $N$,
-%thus $\hat{A}\simeq\{1,\dots, N\}$. We label equivalence classes by
-%$\textbf{n}_i$, then $\hat{A}\simeq\{\textbf{n}_1,\dots,\textbf{n}_N\}$.
-%\newline
-%
-%\textbf{Hilbert Space:} Since every Hilbert space that acts faithfully on a
-%C* algebra is completely reducible, it is isomorphic to the composition
-%of irreducible representations. $H \simeq \bigoplus_{i=1}^N\mathbb{C}^{n_i}
-%\otimes V_i$. Where all $V_i$'s are Vector spaces, their dimension is the
-%multiplicity of the representation landed by $\textbf{n}_i$ to $V_i$ itself
-%by the multiplicity space.
-%\newline
-%
-%\textbf{Finite Dirac Operator:} $D_{ij}$ is connecting nodes $\textbf{n}_i$
-%and $\textbf{n}_j$, with a symmetric map $D_{ij}:\mathbb{C}^{n_i}\otimes V_i
-%\rightarrow \mathbb{C}^{n_j}\otimes V_j$
-%\newline
-%
-%To draw a graph, draw nodes in position $\textbf{n}_i\in \hat{A}$.
-%Multiple nodes at the same position represent multiplicities in $H$.
-%Draw lines between nodes to represent $D_{ij}$.
-%
-%\begin{figure}[h!]
-% \centering
-%\begin{tikzpicture}
-% \node[draw, label=above:{$\textbf{n}_1$},circle, thick] at (-3,0) {};
-% \node[label=above:{$\dots$}] at (-2,0) {};
-% \node[draw, label=above:{$\textbf{n}_i$},circle, thick] at (-1,0) {};
-% \node[label=above:{$\dots$}] at (0,0) {};
-% \node[draw, label=above:{$\textbf{n}_j$},circle, thick] at (1,0) {};
-% \node[draw, label=above:{},circle, thick, inner sep=0cm, minimum
-% size=0.2cm] at (1,0) {};
-% \node[label=above:{$\dots$}] at (2,0) {};
-% \node[draw, label=above:{$\textbf{n}_N$},circle, thick] at (3,0) {};
-%
-% \draw[style=thick, -] (-1,-0.2) -- (1,-0.2);
-% \draw[style=thick, -] (-1,0.2) -- (1,0.2);
-% \path[style=thick, -] (-1,-0.2) edge[bend right=15]
-% node[pos=0.5,below] {} (3,-0.2);
-% \end{tikzpicture}
-% \caption{Example}
-%\end{figure}
-%
-%\subsection{Finite Real Noncommutative Spaces}
-%\subsubsection{Finite Real Spectral Triples}
-%Add on to finite real spectral triples a \textit{real structure}. The
-%requirement is that $H$ is a $A$-$A$-bimodule (before only a $A$-left
-%module).
-%\newline
-%
-%For this we introduce a $\mathbb{Z}_2$-grading $\gamma$ with
-%\begin{align}
-% &\gamma ^* = \gamma \\
-% &\gamma ^2 = 1 \\
-% &\gamma D = - D \gamma\\
-% &\gamma a = a \gamma \;\;\;\; a\in A
-%\end{align}
-%
-%\begin{definition}
-% A \textit{finite real spectral triple} is given by a finite spectral
-% triple $(A, H, D)$ and a anti-unitary operator $J:H\rightarrow H$ called
-% the \textit{real structure}, such that
-% \begin{align}
-% a^\circ := J a^* J^{-1}
-% \end{align}
-% is a right representation of $A$ on $H$, that is $(ab)^\circ = b^\circ
-% a^\circ$. With two requirements
-% \begin{align}
-% &[a, b^\circ] = 0\\
-% &[[D, a],b^\circ] = 0.
-% \end{align}
-% They are called the \textit{commutant property}, and mean that the left
-% action of an element in $A$ and $\Omega _D^1(A)$ commutes with the right
-% action on $A$.
-%\end{definition}
-%\begin{definition}
-% The $KO$-dimension of a real spectral triple is determined by the sings
-% $\epsilon, \epsilon ' ,\epsilon '' \in \{-1, 1\}$ appearing in
-% \begin{align}
-% &J^2 = \epsilon \\
-% &JD = \epsilon \ DJ\\
-% &J\gamma = \epsilon '' \gamma J.
-% \end{align}
-%\end{definition}
-%\begin{table}[h!]
-% \centering
-% \caption{$KO$-dimension $k$ modulo $8$ of a real spectral triple}
-% \begin{tabular}{ c | c c c c c c c c}
-% \hline
-% $k$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
-% \hline
-% $\epsilon$ & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
-% $\epsilon '$ & 1 & -1 & 1 & 1 & 1 & -1 & 1 & 1 \\
-% $\epsilon ''$ & 1 & & -1 & & 1 & & -1 & \\
-% \hline
-% \end{tabular}
-%\end{table}
-%
-%
-%\begin{definition}
-%An opposite-algebra $A^\circ$ of a $A$ is defined to be equal to $A$ as a
-%vector space with the opposite product
-%\begin{align}
-% &a\circ b := ba\\
-% &\Rightarrow a^\circ = Ja^* J^{-1} \;\;\; \text{defines the left
-% representation of $A^\circ$ on $H$}
-%\end{align}
-%\end{definition}
-%
-%
-%\begin{example}
-% Matrix algebra $M_N(\mathbb{C})$ acting on $H=M_N(\mathbb{C})$ by left
-% matrix multiplication with the Hilbert Schmidt inner product.
-% \begin{align}
-% \langle a , b \rangle = \text{Tr}(a^* b)
-% \end{align}
-% Then we define $\gamma (a) = a$ and $J(a) = a^*$ with $a\in H$.
-% Since $D$ mus be odd with respect to $\gamma$ it vanishes identically.
-%\end{example}
-%
-%\begin{definition}
-% We call $\xi \in H$ \textbf{cyclic vector} in $A$ if:
-% \begin{align}
-% A\xi := { a\xi:\;\; a\in A} = H
-% \end{align}
-%
-% We call $\xi \in H$ \textbf{separating vector} in $A$ if:
-% \begin{align}
-% a\xi = 0\;\; \Rightarrow \;\; a=0;\;\;\; a\in A
-% \end{align}
-%\end{definition}
-%
-%%\begin{MyExercise}
-%% \textbf{
-%% In the previous example, show that the right action on $M_N(\mathbb{C})$
-%% on $H = M_N(\mathbb{C})$ as defined by $a \mapsto a^\circ$
-%% is given by right matrix multiplication.
-%%}\newline
-%%
-%% \begin{align}
-%% a^\circ \xi = J a^* J^{-1}\xi = Ja^* \xi^* = J\xi a=\xi^* a
-%% \end{align}
-%%\end{MyExercise}
-%%\begin{MyExercise}
-%% \textbf{
-%% Let $A= \bigoplus _i M_{n_i}(\mathbb{C})$, represented on $H = \bigoplus_i \mathbb{C}^{n_i}
-%% \otimes \mathbb{C}^{m_i}$, meaning that the irreducible representation $\textbf{n}_i$ has
-%% multiplicity $m_i$.
-%% \begin{enumerate}
-%% \item Show that the commutant $A'$ of $A$ is $A'\simeq \bigoplus_i M_{m_i} (\mathbb{C})$. As a consequence show $A'' \simeq A$.
-%% \item Show that if $\xi$ is a separating vector for $A$ than it is cyclic for $A'$.
-%% \end{enumerate}
-%% }
-%%
-%%
-%% \begin{enumerate}
-%% \item We know the multiplicity space is $V_i = \mathbb{C}^{m_i}$. We know that
-%% for $T\in H$ and
-%% $a\in A'$ to work we need $aT=Ta$ by laws of matrix multiplication we need
-%% $A' \simeq \oplus _i M_{m_i}(\mathbb{C})$ for this to work since $H = \bigoplus_i
-%% \mathbb{C}^{n_i}
-%% \otimes \mathbb{C}^{m_i}$
-%%
-%% \item Suppose $\xi$ is cyclic for $A$ then $A'\xi = \{0\}$. Under the action of $A$ we
-%% then have $A'A\xi = AA' \xi = 0 \Rightarrow A' = 0$.\\
-%% Suppose now $\xi$ is separating for $A'$, we have $A'\xi = \{0\}$. We can define a
-%% projection in $A'$, $A\xi = P'$. With this projection we have $(1-P')\xi = 0
-%% \Rightarrow 1-P' = 0 \Rightarrow A\xi = H$.
-%% \end{enumerate}
-%%\end{MyExercise}
-%%\begin{MyExercise}
-%% \textbf{ Suppose $(A, H, D = 0)$ is a finite spectral triple such that $H$ possesses a
-%% cyclic and separating vector for $A$.
-%% \begin{enumerate}
-%% \item Show that the formula $S(a \xi) = a* \xi$ defines a anti-linear operator\\
-%% $S: H \rightarrow H$.
-%% \item Show that $S$ is invertible
-%% \item Let $J: H \rightarrow H$ be the operator in $S = J \Delta ^{1/2}$ with
-%% $\Delta = S^*S$. Show that $J$ is anti-unitary
-%% \end{enumerate}
-%% }
-%%
-%%
-%% \begin{enumerate}
-%% \item By composition $S(a\xi) = a*\xi$ this is literally anti-linearity. Does this mean
-%% $S\xi = \xi$?
-%% \item Let $\xi \in H$ be cyclic then: $S(A\xi) = A^*\xi = A\xi = H$. The same has to work
-%% for $S^{-1}$ if not then $\xi$ wouldn't exist. $S^{-1}(A^*\xi) = S^{-1}(H) = H$.
-%% \item Since $S$ is bijective then $\Delta ^{1/2}$ and $J$ need to be bijective.
-%% We also have $J = S \Delta^{-1/2}$ and $\Delta^* = \Delta$\\
-%% Now let $\xi _1 , \xi _2 \in H$ \begin{align}
-%% <J \xi _1 , J \xi _2 > &= < J^*J\xi_1 , \xi_2>^* =\\
-%% &= <(\Delta ^{-1/2})^* S^* S \Delta ^{-1/2} \xi_1, \xi_2>^* = \\
-%% &= <(\Delta^{-1/2})^* \Delta \Delta^{-1/2} \xi_1, \xi_2>^* =\\
-%% &= <\Delta^{-1/2} \Delta^{1/2}\Delta^{1/2} \Delta^{-1/2} \xi_1, \xi_2>^* =\\
-%% &= <\xi _1, \xi_2>^* = <\xi_2 , \xi_1>.
-%% \end{align}
-%% \end{enumerate}
-%%\end{MyExercise}
-%\subsubsection{Morphisms Between Finite Real Spectral Triples}
-%Extend unitary equivalence of finite spectral triples to real ones (with $J$
-%and $\gamma$)
-%
-%\begin{definition}
-% We call two finite real spectral triples $(A_1, H_1 ,D_1 ; J_1 , \gamma
-% _1)$ and $(A_2, H_2, D_2; J_2, \gamma _2)$ unitarily equivalent if $A_1 =
-% A_2$ and if there exists a unitary operator $U: H_1 \rightarrow H_2$ such
-% that
-% \begin{align}
-% &U\pi_1(a) U^* = \pi _2(a)\\
-% &UD_1U^*=D_2\\
-% &U\gamma _1 U^* = \gamma _2\\
-% &UJ_1 U^* = J_2
-% \end{align}
-%\end{definition}
-%\begin{definition}
-% Let $E$ be a $B$-$A$ bimodule. The \textit{conjugate Module} $E^\circ$ is
-% given by the $A$-$B$-bimodule.
-% \begin{align}
-% E^\circ = \{\bar{e} : e\in E\}
-% \end{align}
-% with
-% \begin{align}
-% a \cdot \bar{e} \cdot b = b^* \bar{e} a^* \;\;\;\; \forall a\in A, b \in
-% B
-% \end{align}
-%\end{definition}
-%$E^\circ$ is not a Hilbert bimodule for $(A, B)$ because it doesn't have a
-%natural $B$-valued inner product. But there is a $A$-valued inner product on
-%the left $A$-module $E^\circ$ with
-%\begin{align}
-% \langle \bar{e}_1, \bar{e}_2 \rangle = \langle e_2 , e_1 \rangle
-% \;\;\;\; e_1, e_2 \in E
-%\end{align}
-%and linearity in $A$:
-%\begin{align}
-% \langle a \bar{e}_1, \bar{e}_2 \rangle = a \langle \bar{e}_1, \bar{e}_2
-% \rangle \;\;\;\; \forall a \in A.
-%\end{align}
-%
-%%\begin{MyExercise}
-%% \textbf{Show that $E^\circ$ is a Hilbert bimodule $(B^{\circ}, A^{\circ})$
-%% }\newline
-%%
-%%
-%% Straightforward show properties of the Hilbert bimodule and its $B^{\circ}$
-%% valued inner product. Let $\bar{e}_1, \bar{e}_2 \in E^{\circ}$ and $a^\circ \in A,
-%% b^\circ \in B$. \\
-%% \begin{align}
-%% <\bar{e}_1, a^\circ \bar{e}_2> &= <\bar{e}_1, Ja^*J^{-1} \bar{e}_2>=\\
-%% &= <\bar{e}_1 , J a^* e_2> = \\
-%% &= <J^{-1} e_1, a^* e_2> =\\
-%% & = <a^* e_1, e_2>= <J^{-1}(a^\circ)^* J e_1, e_2> = \\
-%% & = <J^{-1} (a^\circ)^* \bar{e}_1, e_2> =\\
-%% & = <(a^\circ)^* \bar{e}_1 , \bar{e}_2>.
-%% \end{align}
-%%
-%% Next $<\bar{e}_1, \bar{e}_2 b^\circ> = <\bar{e}_1, \bar{e_2}> b^\circ$.
-%% \begin{align}
-%% <\bar{e}_1, \bar{e}_2 b^\circ> &= <\bar{e}_1, \bar{e}_2 Jb^*J^{-1}> =\\
-%% &= <\bar{e}_1, \bar{e_2}> Jb^*J^{-1} = \\
-%% &= <\bar{e}_1, \bar{e}_2> b^\circ.
-%% \end{align}
-%% Then:
-%% \begin{align}
-%% (<\bar{e}_1, \bar{e}_2)>_{E^\circ})^* &= (<e_2, e_1>_E)^* =\\
-%% &= <e_1, e_2>_E^* = <\bar{e}_2, \bar{e}_2>_{E^\circ}
-%% \end{align}
-%% And of course $<\bar{e}, \bar{e}> = <e, e> \geq 0$
-%%\end{MyExercise}
-%
-%\subsubsection{Construction of a Finite Real Spectral Triple from a Finite
-%Real Spectral Triple}
-%Given a Hilbert bimodule $E$ for $(B, A)$ we construct a spectral triple
-%$(B, H', D'; J', \gamma ')$ from $(A, H, D; J, \gamma)$
-%
-%For the $H'$ we make a $\mathbb{C}$-valued inner product on $H'$ by combining
-%the $A$ valued inner product on $E$ and $E^\circ$ with the
-%$\mathbb{C}$-valued inner product on $H$.
-%\begin{align}
-% H' := E\otimes _A H \otimes _A E^\circ
-%\end{align}
-%
-%Then the action of $B$ on $H'$ is:
-%\begin{align}
-% b(e_2 \otimes \xi \otimes \bar{e}_2 ) = (be_1) \otimes \xi \otimes
-% \bar{e}_2
-%\end{align}
-%The right action of $B$ on $H'$ defined by action on the right component
-%$E^\circ$
-%\begin{align}
-% J'(e_1 \otimes \xi \otimes \bar{e}_2) = e_2 \otimes J \xi \otimes
-% \bar{e}_1
-%\end{align}
-%with $b^\circ = J' b^* (J')^{-1}$, $b^* \in B$ action on $H'$.
-%\newline
-%
-%
-%\newpage
-%%\begin{MyExercise}
-%% \textbf{ Let $\nabla : E \Rightarrow E \otimes _A \Omega _d^1 (A)$ be a right connection on $E$
-%% consider the following anti-linear map:
-%% \begin{align}
-%% \tau : E \otimes_A \Omega _D^1 (A) &\rightarrow \Omega _D^1 (A) \otimes_A E^\circ\\
-%% e \otimes \omega &\mapsto -\omega ^* \otimes \bar{e}
-%% \end{align}
-%% Show that the map $\bar{\nabla} : E^\circ \rightarrow \Omega _D^1(A) \otimes E^\circ$
-%% with $\bar{\nabla}(\bar{e}) = \tau \circ \nabla(e)$ is a left connection, that means
-%% show that it satisfied the left Leibniz rule:
-%% \begin{equation}
-%% \bar{\nabla}(a\bar{e}) = [D, a] \otimes \bar{e} + a \bar{\nabla}(\bar{e})
-%% \end{equation}
-%% }
-%%
-%% Hagime:
-%% \begin{align}
-%% &\text{For one:}\\
-%% &\tau \circ \nabla(ae) = \bar{\nabla}(a\bar{e}) = \bar{\nabla}(a^* \bar{e})\\
-%% &\text{For two:}\\
-%% &\tau \circ \nabla(ae) = \tau(\nabla(e)a) + \tau \circ(e \otimes d(a))=\\
-%% &=a^*\bar{\nabla}(\bar{e}) - d(a)^* \otimes \bar{e}. \\
-%% &= a^*\bar{\nabla}(\bar{e}) + d(a^*) \otimes \bar{e}.
-%% \end{align}
-%%\end{MyExercise}
-%Then the connections
-%\begin{align}
-% &\nabla: E \rightarrow E\otimes _A \Omega _D ^1(A) \\
-% &\bar{\nabla}:E^\circ \rightarrow \Omega _D^1(A) \otimes _A E^\circ
-%\end{align}
-%give us the Dirac operator on $H' = E \otimes _A H \otimes _A E^\circ$
-%\begin{align}
-% D'(e_1 \otimes \xi \otimes \bar{e}_2) = (\nabla e_1) \xi \otimes
-% \bar{e_2}+ e_1 \otimes D\xi \otimes \bar{e}_2 + e_1 \otimes
-% \xi(\bar{\nabla}\bar{e}_2)
-%\end{align}
-%
-%And the right action of $\omega \in \Omega _D ^1(A)$ on $\xi \in H$ is
-%defined by
-%\begin{align}
-% \xi \mapsto \epsilon' J \omega ^* J^{-1}\xi
-%\end{align}
-%
-%Finally for the grading
-%\begin{align}
-% \gamma ' = 1 \otimes \gamma \otimes 1
-%\end{align}
-%
-%\begin{theorem}
-% Suppose $(A, H, D; J, \gamma)$ is a finite spectral triple of
-% $KO$-dimension $k$, let $\nabla$ be like above satisfying the
-% compatibility condition (like with finite spectral triples).
-%
-% Then $(B, H',D'; J', \gamma')$ is a finite spectral triple of
-% $KO$-Dimension $k$. ($H', D', J', \gamma'$ like above)
-%\end{theorem}
-%
-%\begin{proof}
-% The only thing left is to check if the $KO$-dimension is preserved,
-% for this we check if the $\epsilon$'s are the same.
-% \begin{align}
-% &(J')^2 = 1 \otimes J^2 \otimes 1 = \epsilon\\
-% &J' \gamma '= \epsilon ''\gamma'J'
-% \end{align}
-% and for $\epsilon '$
-% \begin{align}
-% J'D'(e_1 \otimes \xi \otimes \bar{e}_2)&=J'((\nabla e_1) \xi \otimes
-% \bar{e_2} + e_1 \otimes D\xi \otimes \bar{e}_2 + e_1 \otimes \xi (\tau
-% \nabla e_2))\\
-% &= \epsilon' D'(e_2 \otimes J\xi \otimes \bar{e}_2)\\
-% &= \epsilon' D'J'(e_1 \otimes \xi \bar{e}_2)
-% \end{align}
-%\end{proof}
-%
-%
-%\end{document}
+To summarize, there is a duality between finite spaces and Morita equivalence classes of matrix algebras. By replacing $*$-homomorphism $A\rightarrow B$ with Hilbert bimodules $(A,B)$ we introduce
+a richer structure of morphism between matrix algebras.
diff --git a/src/thesis/chapters/finitencg.tex b/src/thesis/chapters/finitencg.tex
@@ -0,0 +1,693 @@
+
+\subsubsection{Metric on Finite Discrete Spaces}
+Let us come back to our finite discrete space $X$, we can describe it by a
+structure space $\hat{A}$ of a matrix algebra $A$. To describe distance between
+two points in $X$ (as we would in a metric space) we use an array $\{d_{ij}\}_{i,
+j \in X}$ of \textit{real non-negative} entries in $X$ such that
+\begin{itemize}
+ \item $d_{ij} = d_{ji}$ Symmetric
+ \item $d_{ij} \leq d_{ik} d_{kj}$ Triangle Inequality
+ \item $d_{ij} = 0$ for $i=j$ (the same element)
+\end{itemize}
+
+In the commutative case, the algebra $A$ is commutative and can describe the
+metric on $X$ in terms of algebraic data.
+\begin{theorem}
+ Let $d_{ij}$ be a metric on $X$ a finite discrete space with $N$ points, $A = \mathbb{C}^N$
+ with elements $a = (a(i))_{i=1}^N$ such that $\hat{A} \simeq X$. Then there exists a
+ representation $\pi$ of $A$ on a finite-dimensional inner product space $H$ and a symmetric
+ operator $D$ on $H$ such that
+ \begin{equation}
+ d_{ij} = \sup_{a\in A}\bigg\{\big|a(i)-a(j)\big| : |\big|\big[D,
+ \pi(a)]\big|\big| \leq 1\bigg\}
+ \end{equation}
+\end{theorem}
+
+\begin{proof}
+ We claim that this would follow from the equality:
+ \begin{equation}
+ \big|\big|[D, \pi(a)]\big|\big| = \max_{k\neq l}
+ \bigg\{\frac{1}{d_{kl}}\big|a(k) - a(l)\big|\bigg\}
+ \label{induction}
+ \end{equation}
+ This can be proved with induction. Set $N=2$ then $H=\mathbb{C}^2$, $\pi:A\rightarrow L(H)$ and
+ a hermitian matrix $D$.
+ \begin{align}
+ \pi(a) =
+ \begin{pmatrix}
+ a(1) & 0 \\
+ 0 & a(2)
+ \end{pmatrix}
+ \;\;\;\;
+ D =
+ \begin{pmatrix}
+ 0 & (d_{12})^{-1} \\
+ (d_{21})^{-1} & 0
+ \end{pmatrix}
+ \end{align}
+ Then we commpute the commutator
+ \begin{align}
+ \big|\big|[D, \pi(a)]\big|\big| = (d_{12})^{-1} \big| a(1) - a(2)\big|
+ \end{align}
+
+ For the case $A=\mathbb{C}^3$, we have $H = (\mathbb{C}^2)^{\oplus 3} = H_2
+ \oplus H_2^1 \oplus H_2^2$. The the representation $\pi (a)$ reads
+ \begin{align}
+ \pi((a(1), a(2), a(3)) &=
+ \begin{pmatrix}
+ a(1) & 0 \\ 0 & a(2)
+ \end{pmatrix} \oplus
+ \begin{pmatrix}
+ a(1) & 0 \\ 0 & a(3)
+ \end{pmatrix} \oplus
+ \begin{pmatrix}
+ a(2) & 0 \\ 0 & a(2)
+ \end{pmatrix} \nonumber \\
+ & = \text{diag}\big(a(1), a(2), a(1), a(3), a(2),
+ a(3)\big)
+ \end{align}
+ And the operator $D$ takes the form
+ \begin{align}
+ D &=
+ \begin{pmatrix}
+ 0 & x_1 \\ x_1 & 0
+ \end{pmatrix} \oplus
+ \begin{pmatrix}
+ 0 & x_2 \\ x_2 & 0
+ \end{pmatrix} \oplus
+ \begin{pmatrix}
+ 0 & x_3 \\ x_3 & 0
+ \end{pmatrix} \nonumber \\
+ &=
+ \begin{pmatrix}
+ 0 & x_1 & 0 & 0 & 0 & 0 \\
+ x_1 & 0 & 0 & 0 & 0 & 0 \\
+ 0 & 0 & 0 & x_2 & 0 & 0 \\
+ 0 & 0 & x_2 & 0 & 0 & 0 \\
+ 0 & 0 & 0 & 0 & 0 & x_3 \\
+ 0 & 0 & 0 & 0 & x_3 & 0 \\
+ \end{pmatrix}.
+ \end{align}
+ Then the norm of the commutator would be the largest eigenvalue
+ \begin{align}\label{eq:skew matrix}
+ &\big|\big|[D, \pi(a)]\big|\big| = \big|\big|D\pi(a) - \pi(a)D\big|\big|,
+ \end{align}
+ where the matrix in the norm from equation \eqref{eq:skew matrix} is a
+ skew symmetric matrix. Its eigenvalues are $i\lambda_1, i\lambda_2,
+ i\lambda_3, i\lambda_4$. The $\lambda$'s are on the upper and lower
+ diagonal. The matrix norm would be the maximum of the norm with the
+ larges eigenvalues:
+ \begin{align}
+ \big|\big|[D, \pi(a)]\big|\big| = \max_{a\in A}\bigg\{&x_1\big|a(2)-a(1)\big|,\nonumber\\ &x_2\big|(a(3)-a(1))\big|,\nonumber\\
+ &x_3\big|(a(3)-a(2))\big|\bigg\}.
+ \end{align}
+ Hence the metric turns out to be
+ \begin{align}
+ d =
+ \begin{pmatrix}
+ 0 & a(1)-a(2) & a(1)-a(3)\\
+ a(2)-a(1) & 0 & a(2)-a(3)\\
+ a(3)-a(1) & a(3)-a(2) & 0
+ \end{pmatrix}
+ \end{align}
+
+ Suppose this holds for $N$ with $\pi_N$, $H_N = \mathbb{C}^N$ and $D_N$.
+ Then it has to holds for $N+1$ with $H_{N+1} = H_{N} \oplus \bigoplus_{i=1}^N
+ H_N^i$, since the representation reads
+ \begin{align}
+ \pi_{N+1}(a(1),\dots,a(N+1)) &= \pi_N(a(1),\dots,a(N))
+ \oplus
+ \begin{pmatrix}
+ a(1) & 0 \\
+ 0 & a(N+1)
+ \end{pmatrix} \oplus \nonumber\\
+ &\oplus \cdots \oplus
+ \begin{pmatrix}
+ a(N) & 0 \\
+ 0 1 & a(N+1)
+ \end{pmatrix}
+ \end{align}
+ And the operator $D_{N+1}$ is
+ \begin{align}
+ D_{N+1} &= D_N
+ \oplus
+ \begin{pmatrix}
+ 0 & (d_{1(N+1)})^{-1} \\
+ (d_{1(N+1)})^{-1} & 0
+ \end{pmatrix}\oplus \nonumber \\
+ &\oplus \cdots \oplus
+ \begin{pmatrix}
+ 0 & (d_{N(N+1)})^{-1} \\
+ (d_{N(N+1)})^{-1} & 0
+ \end{pmatrix}
+ \end{align}
+ From this follows equation \eqref{induction}.
+ Thus we can continue the proof by setting for fixed $i, j$, $a(k) =
+ d_{ik}$, which then gives $|a(i) - a(j)| = d_{ij}$ and thereby it follows
+ that
+ \begin{align}
+ \frac{1}{d_{kl}} \big| a(k) - a(l) \big| = \frac{1}{d_{kl}} \big|
+ d_{ik} - d_{il} \big| \leq 1.
+ \end{align}
+\end{proof}
+
+%---------------- EXERCISE
+To get a better understanding of the results of the theorem let us compute a
+metric on the space of three points given by $d_{ij} = \sup_{a\in A}\{|a(i) -
+a(j)|: ||[D, \pi(a)]|| \leq 1\}$ for the set of data $A = \mathbb{C}^3$ acting
+in the defining representation $H = \mathbb{C}^3$, and
+\begin{align}
+ D =
+ \begin{pmatrix}
+ 0 & d^{-1} & 0 \\ d^{-1} & 0 & 0 \\ 0 & 0 & 0
+ \end{pmatrix},
+\end{align}
+for some $d \in \mathbb{R}$.
+From the data $A=\mathbb{C}^3$, $H=\mathbb{C}^3$ and $D$ we compute the
+commutator
+\begin{align}
+ \big|\big|[D, \pi(a)]\big|\big| &= d^{-1}\left|\left|
+\begin{pmatrix}
+ 0 & a(2)-a(1) & 0 \\
+ -(a(2)-a(1)) & 0 & 0 \\
+ 0 & 0 & 0
+\end{pmatrix} \right|\right|.
+\end{align}
+Hence the metric is
+\begin{align}
+d =
+ \begin{pmatrix}
+ 0 & a(1)-a(2) & a(1) \\
+ a(2)-a(1) & 0 & a(2) \\
+ -a(1) & -a(2) & 0
+ \end{pmatrix}.
+\end{align}
+%---------------- EXERCISE
+
+The translation of the metric on $X$ into algebraic data assumes commutativity
+in $A$, this can be extended to a noncommutative matrix algebra, by the
+following metric on a structure space $\hat{A}$ of a matrix algebra
+$M_{n_i}(\mathbb{C}$
+\begin{equation}
+ d_{ij} = \sup_{a\in A}\big\{|\text{Tr}(a(i)) - \text{Tr}((a(j))|: ||[D,
+ a]|| \leq 1\big\}.\label{eq:discretemetric}
+\end{equation}
+Equation \eqref{eq:discretemetric} is special case of the Connes' distance
+formula on a structure space of $A$.
+
+Finally we have all three ingredients to define a finite spectral triple, an
+mathematical structure which encodes finite discrete geometry into algebraic data.
+\begin{definition}
+ A \textit{finite spectral triple} is a tripe $(A, H, D)$, where $A$ is a unital $*$-algebra,
+ faithfully represented on a finite-dimensional Hilbert space $H$, with a symmetric operator
+ $D: H \rightarrow H$. (Note that $A$ is automatically a matrix algebra.)
+\end{definition}
+
+\subsubsection{Properties of Matrix Algebras}
+\begin{lemma}
+ If $A$ is a unital C* algebra acting faithfully on a finite
+ dimensional Hilbert space, then $A$ is a matrix algebra of the Form:
+ \begin{align}
+ A \simeq \bigoplus _{i=1}^N M_{n_i}(\mathbb{C})
+ \end{align}
+\end{lemma}
+\begin{proof}
+ The wording 'acting faithfully on a Hilbertspace' means that the
+ $*$-representation is injective, or for a $*$-homomorphism that means
+ one-to-one correspondance. And since $A$ acts faithfully on a Hilbert
+ space, this means that $A$ is a $*$ subalgebra of a matrix algebra $L(H) = M_{\dim
+ (H)}(\mathbb{C}$. Hence it follows, that $A$ is isomorphic to a matrix
+ algebra.
+\end{proof}
+
+A simple ilustration would be for an algebra $A = M_n(\mathbb{C})$ and
+$H=\mathbb{C}^n$. Since $A$ acts on $H$ with matrix multiplication and standard
+inner product and $D$ on $H$ is a hermitian matrix $n\times n$ matrix.
+
+\begin{definition}
+ Given an finite spectral triple $(A, H, D)$, the $A$-bimodule of
+ Connes' differential one-forms is
+ \begin{align}\label{eq:connesoneforms}
+ \Omega _D ^1 (A) := \left\{ \sum _k a_k[D, b_k]: a_k, b_k \in A
+ \right\}.
+ \end{align}
+\end{definition}
+Then there is a map $d:A\rightarrow \Omega _D ^1 (A)$, $d = [D, \cdot]$. Where
+$d$ is a derivation of the $*$-algebra in the sence that
+\begin{align}
+ d(ab) = d(a)b + ad(b), \\
+ d(a^*) = -d(a)^*.
+\end{align}
+Since we have $d(\cdot) = [D, \cdot]$, we can easily check the above equations
+\begin{align}
+ d(a\ b) &= [D, a\ b] = [D, a]b + a[D,b]\nonumber\\
+ &= d(a)\ b + a\ d(b)
+\end{align}
+And
+\begin{align}
+ d(a^*) &= [D, a^*] = Da^* - a^*D \nonumber\\
+ &=-(D^*\ a - a\ D^*) = -[D^*, a] \nonumber\\
+ &= -d(a)^*.
+\end{align}
+Furthermore $\Omega _D^1 (A)$ is an $A$-bimodule, which can be seen by
+rewriting the definition \eqref{eq:connesoneforms} into
+\begin{align}
+ a\ (a_k[D, b_k])\ b &= a\ a_k(D\ b_k - b_k\ D)\ b = \nonumber\\
+ &= a\ a_k(D\ b_k\ b - b_k\ D\ b)=\nonumber\\
+ &= a\ a_k(D\ b_k\ b - b_k\ D\ b - b_k\
+ b\ D +b_k\ b\ D)=
+ \nonumber\\
+ &= a\ a_k(D\ b_k\ b-b_k\ b\ D + b_k\ b\ D - b_k\ D\ b) = \nonumber \\
+ &= a\ a_k [D, b_k\ b] + a\ a_k\ b [D, b]=\nonumber\\
+ &= \sum _k\ a_k'\ [D, b_k']
+\end{align}
+
+\begin{lemma}
+ Let $\big(A, H, D\big) = \big(M_n(\mathbb{C}), \mathbb{C}^n, D\big)$, where
+ $D$ is a hermitian $n\times n$ matrix. If $D$ is not a multiple of the
+ identity then
+ \begin{align}
+ \Omega _D ^1 (A) \simeq M_n(\mathbb{C}) = A
+ \end{align}
+\end{lemma}
+\begin{proof}
+ Assume $D = \sum _i \lambda _i e_{ii}$ is diagonal, $\lambda _i \in \mathbb{R}$ and
+ $\{e_{ij}\}$ is the basis of $M_n(\mathbb{C})$. Then for fixed $i$, $j$ choose $k$
+ such that $\lambda _k \neq \lambda _j$, hence we have
+ \begin{align} \label{eq:basis}
+ \left(\frac{1}{\lambda _k - \lambda _j} e_{ik}\right) [D, e_{kj}] =
+ e_{ij},
+ \end{align}
+ for $e_{ij}\in \Omega _D ^1 (A)$ by the above definition
+ \eqref{eq:connesoneforms}. Ultimately we have
+ \begin{align}
+ \Omega _D ^1
+ (A) \subset L(\mathbb{C}^n) = H \simeq M_n(\mathbb{C}) = A
+ \end{align}
+\end{proof}
+
+ Consider an example
+ \begin{align}
+ \left(A=\mathbb{C}^2, H=\mathbb{C}^2,
+ D = \begin{pmatrix} 0 & \lambda \\ \bar{\lambda} & 0
+ \end{pmatrix}\right)
+ \end{align}
+ with $\lambda \neq 0$. We can show that $\Omega _D^1(A)
+ \simeq M_2(\mathbb{C})$. The Hilbert Basis $D$ can be extended in terms of
+ the basis of $M_2(\mathbb{C})$, plugging this into Equation
+ \eqref{eq:basis} will get us the same cyclic result, thus
+ $\Omega _D^1(A) \simeq M_2(\mathbb{C})$.
+\
+
+\subsubsection{Morphisms Between Finite Spectral Triples}
+\begin{definition}
+ Two finite spectral tripes $(A_1, H_1, D_1)$ and $(A_2, H_2, D_2)$ are
+ called unitarily equivalent if
+ \begin{itemize}
+ \item $A_1 = A_2$
+ \item $\exists \;\; U: H_1 \rightarrow H_2$, unitary with
+ \begin{enumerate}
+ \item $U\pi_1(a)U^* = \pi_2(a)$ with $a \in A_1$
+ \item $UD_1 U^* = D_2$
+ \end{enumerate}
+ \end{itemize}
+\end{definition}
+
+Some remarks
+\begin{itemize}
+ \item the above is an equivalence relation
+ \item spectral unitary equivalence is given by the unitaries of the
+ matrix algebra itself
+ \item for any such $U$ then $(A, H, D) \sim (A, H, UDU^*)$
+ \item $UDU^* = D + U[D, U^*]$ of the form of elements in
+ $\Omega _D^1 (A)$.
+\end{itemize}
+
+%\begin{MyExercise}
+% \textbf{
+% Show that the unitary equivalence between finite spectral
+% triples is a equivalence relation
+%}\newline
+%
+% An equivalence relation needs to satisfy reflexivity, symmetry
+% transitivity.
+% Let $(A_1, H_1, D_1)$, $(A_2, H_2, D_2)$ and $(A_3, H_3, D_3)$
+% be three finite spectral triples.
+% \newline
+%
+% For reflexivity $(A_1, H_1, D_1) \sim (A_1, H_1, D_1)$. So there
+% exists a $U: H_1 \rightarrow H_1$ unitary, which is the identity
+% and always exists.
+% \newline
+%
+% For symmetry we need
+% \begin{align}
+% (A_1, H_1, D_1) \sim (A_2, H_2, D_2) \Leftrightarrow
+% (A_2, H_2, D_2) \sim (A_1, H_1, D_1)
+% \end{align}
+% because $U$ is unitary:
+% \begin{align}
+% &U\pi_1(a)U^* = \pi_2(a) \;\;\; | \cdot U^*\boxdot U \\
+% &U^*U\pi_1(a)U^*U = \pi_1(a) = U^*\pi_2(a)U \\
+% \end{align}
+% The same with the symmetric operator $D$.
+% \newline
+%
+% For transitivity we need
+% \begin{align}
+% (A_1, H_1, D_1) &\sim (A_2, H_2, D_2) \;\;\; \text{and} \;\;\;
+% (A_2, H_2, D_2) \sim (A_3, H_3, D_3) \\
+% &\Rightarrow (A_1, H_1, D_1) \sim (A_3, H_3, D_3)
+% \end{align}
+% There are two unitary maps $U_{12}:H_1 \rightarrow H_2$ and
+% $U_{23}: H_2 \rightarrow H_3$ then
+% \begin{align}
+% U_{23}U_{12} \pi_1(a) U^*_{12}U^*_{23} &= U_{23}
+% \pi_2(a) U_23^* \\
+% &= \pi_3(a) \\
+% U_{23}U_{12} D_1U^*_{12}U^*_{23} &= U_{23}
+% D_2 U_23^* \\
+% &= D_3
+% \end{align}
+%\end{MyExercise}
+
+Extending the this relation we look again at the notion of equivalence from
+Morita equivalence of Matrix Algebras.
+\newline
+
+\begin{definition}
+ Let $A$ be an algebra. We say that $I \subset A$, as a vector space, is a
+ right(left) ideal if $ab \in I$ for $a \in A$ and $b\in I$ (or $ba \in
+ I$, $b\in I$, $a\in A$). We call a left-right ideal simply an ideal.
+\end{definition}
+
+Given a Hilbert bimodule $E \in KK_f(B, A)$ and $(A, H, D)$ we construct
+a finite spectral triple on $B$, $(B, H', D')$
+\begin{equation}
+ H' = E \otimes _A H
+\end{equation}
+We might define $D'$ with $D'(e \otimes \xi) = e\otimes D\xi$, thought this
+would not satisfy the ideal defining the balanced tensor product over $A$,
+which is generated by elements of the form
+\begin{align}
+ e a \otimes \xi - e\otimes a \xi ;\;\;\;\; e\in E, a\in A, \xi \in H
+\end{align}
+This inherits the left action on $B$ from $E$ and has a $\mathbb{C}$
+valued inner product space. $B$ also satisfies the ideal.
+\begin{equation}
+ D'(e\otimes \xi) = e \otimes D \xi + \nabla (e) \xi \;\;\;\; e\in
+ E, a\in A
+\end{equation}
+Where $\nabla$ is called the \textit{connection on the right A-module E}
+associated with the derivation $d=[D, \cdot]$ and satisfying the
+\textit{Leibnitz Rule} which is
+\begin{equation}
+ \nabla(ae) = \nabla(e)a + e \otimes [D, a] \;\;\;\;\; e\in E,\; a\in A
+\end{equation}
+Then $D'$ is well defined on $E \otimes _A H$:
+\begin{align}
+ D'(ea \otimes \xi - e \otimes a \xi) &= D'(ea \otimes \xi) - D'(e
+ \otimes \xi) \\
+ &= ea\otimes D\xi + \nabla(ae) \xi - e \otimes D(a\xi ) - \nabla (e)a
+ \xi \\
+ &= 0.
+\end{align}
+With the information thus far we can prove the following theorem
+\begin{theorem}
+ If $(A, H, D)$ a finite spectral triple, $E \in KK_f(B, A)$.
+ Then $(V, E\otimes _A H, D')$ is a finite spectral triple, provided that
+ $\nabla$ satisfies the compatibility condition
+ \begin{equation}
+ \langle e_1, \nabla e_2 \rangle _E - \langle \nabla e_1, e_2
+ \rangle _E = d\langle e_1, e_2 \rangle _E \;\;\;\; e_1, e_2 \in E
+ \end{equation}
+\end{theorem}
+\begin{proof}
+ $E\otimes _A H$ was shown in the previous subsection (text before the
+ theorem). The only thing left is to show that $D'$ is a symmetric
+ operator, this we can just compute. Let $e_1, e_2 \in E$ and $\xi _1,
+ \xi _2 \in H$ then
+ \begin{align}
+ \langle e_1 \otimes \xi _1, D'(e_2 \otimes \xi_2)\rangle _{E\otimes _A H} &=
+ \langle \xi _1, \langle e_1, \nabla e_2\rangle _E \xi _2\rangle + \langle \xi _1 , \langle e_1, e_2\rangle _E D\xi
+ _2\rangle _H \\
+ &= \langle \xi _1, \langle \nabla e_1, e_2\rangle _E \xi _2\rangle _H + \langle \xi _1, d\langle e_1, e_2\rangle _E
+ \xi _2\rangle _H \\
+ &+ \langle D\xi _1,\langle e_1, e_2\rangle _E \xi _2\rangle _H - \langle \xi _1, [D, \langle e_1, e_2\rangle _E] \xi
+ _2 \rangle _H \\
+ &= \langle D'(e_1 \otimes \xi _1), e_2 \otimes \xi _2\rangle _{E \otimes _A H}
+ \end{align}
+\end{proof}
+
+%\begin{MyExercise}
+% \textbf{
+% Let $\nabla$ and $\nabla'$ be two connections on a right $A$-module
+% $E$. Show that $\nabla - \nabla'$ is a right $A$-linear map
+% $E \rightarrow E\otimes _A \Omega _D^1(A)$
+%}\newline
+%
+% Both $\nabla$ and $\nabla'$ need to satisfy the Leiblitz rule, so
+% let's see if $\nabla - \nabla'$ does.
+%
+% \begin{align}
+% \nabla(ea)-\nabla'(ea)&=\nabla(e) + e\otimes[D, a]\\
+% &-(\nabla'(e)a + e\otimes[D',a])\\
+% &=\bar{\nabla}a + e\otimes(Da-aD-D'a+aD')\\
+% &=\bar{\nabla}a + e\otimes((D-D')a-a(D-D'))\\
+% &=\bar{\nabla}a + e\otimes[D', a]\\
+% &=\bar{\nabla}(ea)
+% \end{align}
+% Therefore $\nabla-\nabla'$ is a linear map.
+%\end{MyExercise}
+
+%\begin{MyExercise}
+% \textbf{
+% Construct a finite spectral triple $(A, H', D')$ from $(A, H, D)$
+% \begin{enumerate}
+% \item show that the derivation $d(\cdot):A \rightarrow A\otimes _A
+% \Omega_D^1(A)=\Omega_D^1(A)$ is a connection on $A$
+% considered a right $A$-module
+% \item Upon identifying $A\otimes_A H\simeq H$, what is $D'$
+% when the connection is $d(\cdot)$.
+% \item Use 1) and 2) to show that any connection $\nabla:
+% A\rightarrow A\otimes_A \Omega_D^1(A)$ is given by
+% \begin{align}
+% \nabla = d + \omega
+% \end{align}
+% where $\omega \in \Omega_D^1(A)$
+% \item Upon identifying $A\otimes_A H \simeq H$, what is the
+% difference operator $D'$ with the connection on $A$ given by
+% $\nabla = d + \omega$
+% \end{enumerate}
+%}
+% \begin{enumerate}
+% \item $\nabla(e \cdot a) = d(a)$
+% \item
+% $D'(a\xi) = a(D\xi) + (\nabla a) \xi = D(a\xi)$
+% \item Use the identity element $e \in A$\\
+% $\nabla (e\cdot a) = \nabla(e) a + 1 \otimes d(a)=d(a)
+% \nabla(e) a$
+% \item $D'(a\otimes \xi) = D'(a \xi) = a(D\xi) + (\nabla a)\xi =
+% a(D\xi) + \nabla(e \cdot a) \xi \\
+% = D(a\xi) + \nabla(e) (a\xi)$
+% \end{enumerate}
+%\end{MyExercise}
+
+%\subsubsection{Graphing Finite Spectral Triples}
+%\begin{definition}
+% A \textit{graph} is a ordered pair $(\Gamma ^{(0)}, \Gamma ^{(1)})$.
+% Where $\Gamma ^{(0)}$ is the set of vertices (nodes) and $\Gamma ^{(1)}$
+% a set of pairs of vertices (edges)
+%\end{definition}
+%\begin{figure}[h!]
+% \centering
+%\begin{tikzpicture}[
+% mass/.style = {draw,circle, minimum size=0.2cm, inner sep=0pt, thick},
+% spring/.style = {decorate,decoration={zigzag, pre length=1cm,post length=1cm,segment length=5pt}},]
+% \node[mass] (m1) at (1,1.5) {};
+% \node[mass] (m2) at (-1,1.5) {};
+% \node[mass] (m3) at (0,0) {};
+%
+% \draw (m1) -- (m2);
+% \draw (m1) -- (m3);
+% \draw (m2) -- (m3);
+% \end{tikzpicture}
+% \caption{A simple graph with three vertices and three edges}
+%\end{figure}
+%%\begin{MyExercise}
+%% \textbf{
+%% Show that any finite-dimensional faithful representation $H$ of a matrix
+%% algebra $A$ is completely reducible. To do that show that the complement
+%% $W^{\perp}$ of an $A$-submodule $W\subset H$ is also an $A$-submodule
+%% of $H$.
+%%}\newline
+%%
+%% $A\simeq \bigoplus_{i=1}^N M_{n_i}(\mathbb{C})$ is the matrix algebra
+%% then $H$ is a Hilbert $A$-bimodule and $W$ a submodule of $A$.
+%% Because we have $H = W \cup W^{\perp}$, then $W^{\perp}$ is naturally a
+%% $A$-submodule, because elements in $W^{\perp}$ need to satisfy the
+%% bimodularity.
+%%\end{MyExercise}
+%\begin{definition}
+% A $\Lambda$-decorated graph is given by an ordered pair $(\Gamma,
+% \Lambda)$ of a finite graph $\Gamma$ and a set of positive integers
+% $\Lambda$ with the labeling
+% \begin{itemize}
+% \item of the vetices $v\in \Gamma ^{(0)}$ given by $n(\nu) \in
+% \Lambda$
+% \item of the edges $e = (\nu _1, \nu _2) \in \Gamma ^{(1)}$ by
+% operators
+% \begin{itemize}
+% \item $D_e: \mathbb{C}^{n(\nu _1)} \rightarrow
+% \mathbb{C}^{n(\nu _2)}$
+% \item and $D_e^*: \mathbb{C}^{n(\nu _2)} \rightarrow
+% \mathbb{C}^{n(\nu _1)}$ its conjugate traspose
+% (pullback?)
+% \end{itemize}
+% \end{itemize}
+% such that
+% \begin{equation}
+% n(\Gamma ^{(0)}) = \Lambda
+% \end{equation}
+%\end{definition}
+%\begin{question}
+% Would then $D_e$ be the pullback?
+%\end{question}
+%\begin{question}
+% These graphs are important in the next chapter I should look
+% into it more, I don't understand much here, specific
+% how to construct them with the abstraction of a spectral triple...
+%\end{question}
+%
+%The operator $D_e$ between $\textbf{n}_i$ and $\textbf{n}_j$ add up to
+%$D_{ij}$
+%\begin{align}
+% D_{ij} = \sum\limits_{\substack{e = (\nu _1, \nu _2) \\ n(\nu _1) =
+% \textbf{n}_i \\ n(\nu _2) = \textbf{n}_j}} D_e
+%\end{align}
+%
+%\begin{theorem}
+% There is a on to one correspondence between finite spectral triples
+% modulo unitary equivalence and $\Lambda$-decorated graphs, given by
+% associating a finite spectral triples $(A, H, D)$ to a $\Lambda$ decorated
+% graph $(\Gamma, \Lambda)$ in the following way:
+% \begin{equation}
+% A = \bigoplus _{n\in \Lambda} M_n(\mathbb{C}); \;\;\;
+% H = \bigoplus _{\nu \in \Gamma ^{(0)}} \mathbb{C}^{n(\nu)}; \;\;\;
+% D = \sum _{e \in \Gamma ^{(1)}} D_e + D_e^*
+% \end{equation}
+%\end{theorem}
+% \begin{figure}[h!]
+% \centering
+% \begin{tikzpicture}[
+% mass/.style = {draw,circle, minimum size=0.3cm, inner sep=0pt, thick},
+% ]
+%
+% \node[mass, label={\textbf{n}}] (m1) at (1,0) {};
+% \draw (m1) to [out=330, in=210, looseness=25] node[above] {$D_e$} (m1);
+% \end{tikzpicture}
+% \caption{A $\Lambda$-decorated Graph of $(M_n(\mathbb{C}), \mathbb{C}^n,
+% D = D_e + D_e^*)$}
+%\end{figure}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% Draw a $\Lambda$ decorated graph corresponding to the spectral triple
+%% $(A=\mathbb{C}^3, H=\mathbb{C}^3, D=\begin{pmatrix}0 & \lambda & 0\\
+%% \bar{\lambda} &0 &0 \\ 0&0&0\end{pmatrix})$
+%%}\newline
+%%
+%%\centering
+%%\begin{tikzpicture}[
+%% mass/.style = {draw,circle, minimum size=0.4cm, inner sep=0pt, thick},
+%% spring/.style = {decorate,decoration={zigzag, pre length=1cm,post length=1cm,segment length=5pt}},]
+%% \node[mass] (m1) at (-1,1.5) {\textbf{1}};
+%% \node[mass] (m2) at (1,1.5) {\textbf{2}};
+%% \node[mass] (m3) at (3,1.5) {\textbf{3}};
+%%
+%% \draw[style=thick, -] (1.1,1.7) -- (-1.1,1.7);
+%% \draw[style=thick, -] (1.1,1.3) -- (-1.1,1.3);
+%% \end{tikzpicture}
+%% % \captionof{figure}{Solution}
+%%\end{MyExercise}
+%%\begin{MyExercise}
+%% \textbf{
+%% Use $\Lambda$-decorated graphs to classify all finite spectral triples
+%% (modulo unitary equivalence) on the matrix algebra
+%% $A=\mathbb{C}\oplus M_2(\mathbb{C})$
+%%}\newline
+%%
+%% \centering
+%%\begin{tikzpicture}[
+%% mass/.style = {draw,circle, minimum size=0.4cm, inner sep=0pt, thick},
+%% spring/.style = {decorate,decoration={zigzag, pre length=1cm,post length=1cm,segment length=5pt}},]
+%% \node[mass] (m1) at (-1,1) {\textbf{1}};
+%% \node[mass] (m2) at (1,1) {\textbf{2}};
+%% \node[mass] (m3) at (3,1) {\textbf{3}};
+%%
+%% \node[mass] (m4) at (-1,0) {\textbf{1}};
+%% \node[mass] (m5) at (1,0) {\textbf{2}};
+%% \node[mass] (m6) at (3,0) {\textbf{3}};
+%%
+%% \node[mass] (m7) at (-1,-1) {\textbf{1}};
+%% \node[mass] (m8) at (1,-1) {\textbf{2}};
+%% \node[mass] (m9) at (3,-1) {\textbf{3}};
+%%
+%% \node[mass] (m10) at (-1,-2) {\textbf{1}};
+%% \node[mass] (m11) at (1,-2) {\textbf{2}};
+%% \node[mass] (m12) at (3,-2) {\textbf{3}};
+%%
+%% \draw[style=thick, -] (1.1,0.2) -- (-1.1,0.2);
+%% \draw[style=thick, -] (1.1,-0.2) -- (-1.1,-0.2);
+%% \draw[style=thick, -] (m7) to [out=330, in=210, looseness=10] node[above] {} (m7);
+%% \draw[style=thick, -] (m10) -- (m11) ;
+%%
+%%\end{tikzpicture}
+%%% \captionof{figure}{Solution $A=M_3(\mathbb{C})$}
+%%\end{MyExercise}
+%\subsubsection{Graph Construction of Finite Spectral Triples}
+%\textbf{Algebra:}We know if a acts on a finite dimensional Hilbert space then
+%this C* algebra is isomorphic to a matrix algebra so $A \simeq
+%\bigoplus_{i=1}^{N}M_{n_i}(\mathbb{C})$. Where $i\in
+%\hat{A}$ represents an equivalence class and runs from $1$ to $N$,
+%thus $\hat{A}\simeq\{1,\dots, N\}$. We label equivalence classes by
+%$\textbf{n}_i$, then $\hat{A}\simeq\{\textbf{n}_1,\dots,\textbf{n}_N\}$.
+%\newline
+%
+%\textbf{Hilbert Space:} Since every Hilbert space that acts faithfully on a
+%C* algebra is completely reducible, it is isomorphic to the composition
+%of irreducible representations. $H \simeq \bigoplus_{i=1}^N\mathbb{C}^{n_i}
+%\otimes V_i$. Where all $V_i$'s are Vector spaces, their dimension is the
+%multiplicity of the representation landed by $\textbf{n}_i$ to $V_i$ itself
+%by the multiplicity space.
+%\newline
+%
+%\textbf{Finite Dirac Operator:} $D_{ij}$ is connecting nodes $\textbf{n}_i$
+%and $\textbf{n}_j$, with a symmetric map $D_{ij}:\mathbb{C}^{n_i}\otimes V_i
+%\rightarrow \mathbb{C}^{n_j}\otimes V_j$
+%\newline
+%
+%To draw a graph, draw nodes in position $\textbf{n}_i\in \hat{A}$.
+%Multiple nodes at the same position represent multiplicities in $H$.
+%Draw lines between nodes to represent $D_{ij}$.
+%
+%\begin{figure}[h!]
+% \centering
+%\begin{tikzpicture}
+% \node[draw, label=above:{$\textbf{n}_1$},circle, thick] at (-3,0) {};
+% \node[label=above:{$\dots$}] at (-2,0) {};
+% \node[draw, label=above:{$\textbf{n}_i$},circle, thick] at (-1,0) {};
+% \node[label=above:{$\dots$}] at (0,0) {};
+% \node[draw, label=above:{$\textbf{n}_j$},circle, thick] at (1,0) {};
+% \node[draw, label=above:{},circle, thick, inner sep=0cm, minimum
+% size=0.2cm] at (1,0) {};
+% \node[label=above:{$\dots$}] at (2,0) {};
+% \node[draw, label=above:{$\textbf{n}_N$},circle, thick] at (3,0) {};
+%
+% \draw[style=thick, -] (-1,-0.2) -- (1,-0.2);
+% \draw[style=thick, -] (-1,0.2) -- (1,0.2);
+% \path[style=thick, -] (-1,-0.2) edge[bend right=15]
+% node[pos=0.5,below] {} (3,-0.2);
+% \end{tikzpicture}
+% \caption{Example}
+%\end{figure}
diff --git a/src/thesis/chapters/realncg.tex b/src/thesis/chapters/realncg.tex
@@ -0,0 +1,329 @@
+%\subsection{Finite Real Noncommutative Spaces}
+%\subsubsection{Finite Real Spectral Triples}
+%Add on to finite real spectral triples a \textit{real structure}. The
+%requirement is that $H$ is a $A$-$A$-bimodule (before only a $A$-left
+%module).
+%\newline
+%
+%For this we introduce a $\mathbb{Z}_2$-grading $\gamma$ with
+%\begin{align}
+% &\gamma ^* = \gamma \\
+% &\gamma ^2 = 1 \\
+% &\gamma D = - D \gamma\\
+% &\gamma a = a \gamma \;\;\;\; a\in A
+%\end{align}
+%
+%\begin{definition}
+% A \textit{finite real spectral triple} is given by a finite spectral
+% triple $(A, H, D)$ and a anti-unitary operator $J:H\rightarrow H$ called
+% the \textit{real structure}, such that
+% \begin{align}
+% a^\circ := J a^* J^{-1}
+% \end{align}
+% is a right representation of $A$ on $H$, that is $(ab)^\circ = b^\circ
+% a^\circ$. With two requirements
+% \begin{align}
+% &[a, b^\circ] = 0\\
+% &[[D, a],b^\circ] = 0.
+% \end{align}
+% They are called the \textit{commutant property}, and mean that the left
+% action of an element in $A$ and $\Omega _D^1(A)$ commutes with the right
+% action on $A$.
+%\end{definition}
+%\begin{definition}
+% The $KO$-dimension of a real spectral triple is determined by the sings
+% $\epsilon, \epsilon ' ,\epsilon '' \in \{-1, 1\}$ appearing in
+% \begin{align}
+% &J^2 = \epsilon \\
+% &JD = \epsilon \ DJ\\
+% &J\gamma = \epsilon '' \gamma J.
+% \end{align}
+%\end{definition}
+%\begin{table}[h!]
+% \centering
+% \caption{$KO$-dimension $k$ modulo $8$ of a real spectral triple}
+% \begin{tabular}{ c | c c c c c c c c}
+% \hline
+% $k$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
+% \hline
+% $\epsilon$ & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \\
+% $\epsilon '$ & 1 & -1 & 1 & 1 & 1 & -1 & 1 & 1 \\
+% $\epsilon ''$ & 1 & & -1 & & 1 & & -1 & \\
+% \hline
+% \end{tabular}
+%\end{table}
+%
+%
+%\begin{definition}
+%An opposite-algebra $A^\circ$ of a $A$ is defined to be equal to $A$ as a
+%vector space with the opposite product
+%\begin{align}
+% &a\circ b := ba\\
+% &\Rightarrow a^\circ = Ja^* J^{-1} \;\;\; \text{defines the left
+% representation of $A^\circ$ on $H$}
+%\end{align}
+%\end{definition}
+%
+%
+%\begin{example}
+% Matrix algebra $M_N(\mathbb{C})$ acting on $H=M_N(\mathbb{C})$ by left
+% matrix multiplication with the Hilbert Schmidt inner product.
+% \begin{align}
+% \langle a , b \rangle = \text{Tr}(a^* b)
+% \end{align}
+% Then we define $\gamma (a) = a$ and $J(a) = a^*$ with $a\in H$.
+% Since $D$ mus be odd with respect to $\gamma$ it vanishes identically.
+%\end{example}
+%
+%\begin{definition}
+% We call $\xi \in H$ \textbf{cyclic vector} in $A$ if:
+% \begin{align}
+% A\xi := { a\xi:\;\; a\in A} = H
+% \end{align}
+%
+% We call $\xi \in H$ \textbf{separating vector} in $A$ if:
+% \begin{align}
+% a\xi = 0\;\; \Rightarrow \;\; a=0;\;\;\; a\in A
+% \end{align}
+%\end{definition}
+%
+%%\begin{MyExercise}
+%% \textbf{
+%% In the previous example, show that the right action on $M_N(\mathbb{C})$
+%% on $H = M_N(\mathbb{C})$ as defined by $a \mapsto a^\circ$
+%% is given by right matrix multiplication.
+%%}\newline
+%%
+%% \begin{align}
+%% a^\circ \xi = J a^* J^{-1}\xi = Ja^* \xi^* = J\xi a=\xi^* a
+%% \end{align}
+%%\end{MyExercise}
+%%\begin{MyExercise}
+%% \textbf{
+%% Let $A= \bigoplus _i M_{n_i}(\mathbb{C})$, represented on $H = \bigoplus_i \mathbb{C}^{n_i}
+%% \otimes \mathbb{C}^{m_i}$, meaning that the irreducible representation $\textbf{n}_i$ has
+%% multiplicity $m_i$.
+%% \begin{enumerate}
+%% \item Show that the commutant $A'$ of $A$ is $A'\simeq \bigoplus_i M_{m_i} (\mathbb{C})$. As a consequence show $A'' \simeq A$.
+%% \item Show that if $\xi$ is a separating vector for $A$ than it is cyclic for $A'$.
+%% \end{enumerate}
+%% }
+%%
+%%
+%% \begin{enumerate}
+%% \item We know the multiplicity space is $V_i = \mathbb{C}^{m_i}$. We know that
+%% for $T\in H$ and
+%% $a\in A'$ to work we need $aT=Ta$ by laws of matrix multiplication we need
+%% $A' \simeq \oplus _i M_{m_i}(\mathbb{C})$ for this to work since $H = \bigoplus_i
+%% \mathbb{C}^{n_i}
+%% \otimes \mathbb{C}^{m_i}$
+%%
+%% \item Suppose $\xi$ is cyclic for $A$ then $A'\xi = \{0\}$. Under the action of $A$ we
+%% then have $A'A\xi = AA' \xi = 0 \Rightarrow A' = 0$.\\
+%% Suppose now $\xi$ is separating for $A'$, we have $A'\xi = \{0\}$. We can define a
+%% projection in $A'$, $A\xi = P'$. With this projection we have $(1-P')\xi = 0
+%% \Rightarrow 1-P' = 0 \Rightarrow A\xi = H$.
+%% \end{enumerate}
+%%\end{MyExercise}
+%%\begin{MyExercise}
+%% \textbf{ Suppose $(A, H, D = 0)$ is a finite spectral triple such that $H$ possesses a
+%% cyclic and separating vector for $A$.
+%% \begin{enumerate}
+%% \item Show that the formula $S(a \xi) = a* \xi$ defines a anti-linear operator\\
+%% $S: H \rightarrow H$.
+%% \item Show that $S$ is invertible
+%% \item Let $J: H \rightarrow H$ be the operator in $S = J \Delta ^{1/2}$ with
+%% $\Delta = S^*S$. Show that $J$ is anti-unitary
+%% \end{enumerate}
+%% }
+%%
+%%
+%% \begin{enumerate}
+%% \item By composition $S(a\xi) = a*\xi$ this is literally anti-linearity. Does this mean
+%% $S\xi = \xi$?
+%% \item Let $\xi \in H$ be cyclic then: $S(A\xi) = A^*\xi = A\xi = H$. The same has to work
+%% for $S^{-1}$ if not then $\xi$ wouldn't exist. $S^{-1}(A^*\xi) = S^{-1}(H) = H$.
+%% \item Since $S$ is bijective then $\Delta ^{1/2}$ and $J$ need to be bijective.
+%% We also have $J = S \Delta^{-1/2}$ and $\Delta^* = \Delta$\\
+%% Now let $\xi _1 , \xi _2 \in H$ \begin{align}
+%% <J \xi _1 , J \xi _2 > &= < J^*J\xi_1 , \xi_2>^* =\\
+%% &= <(\Delta ^{-1/2})^* S^* S \Delta ^{-1/2} \xi_1, \xi_2>^* = \\
+%% &= <(\Delta^{-1/2})^* \Delta \Delta^{-1/2} \xi_1, \xi_2>^* =\\
+%% &= <\Delta^{-1/2} \Delta^{1/2}\Delta^{1/2} \Delta^{-1/2} \xi_1, \xi_2>^* =\\
+%% &= <\xi _1, \xi_2>^* = <\xi_2 , \xi_1>.
+%% \end{align}
+%% \end{enumerate}
+%%\end{MyExercise}
+%\subsubsection{Morphisms Between Finite Real Spectral Triples}
+%Extend unitary equivalence of finite spectral triples to real ones (with $J$
+%and $\gamma$)
+%
+%\begin{definition}
+% We call two finite real spectral triples $(A_1, H_1 ,D_1 ; J_1 , \gamma
+% _1)$ and $(A_2, H_2, D_2; J_2, \gamma _2)$ unitarily equivalent if $A_1 =
+% A_2$ and if there exists a unitary operator $U: H_1 \rightarrow H_2$ such
+% that
+% \begin{align}
+% &U\pi_1(a) U^* = \pi _2(a)\\
+% &UD_1U^*=D_2\\
+% &U\gamma _1 U^* = \gamma _2\\
+% &UJ_1 U^* = J_2
+% \end{align}
+%\end{definition}
+%\begin{definition}
+% Let $E$ be a $B$-$A$ bimodule. The \textit{conjugate Module} $E^\circ$ is
+% given by the $A$-$B$-bimodule.
+% \begin{align}
+% E^\circ = \{\bar{e} : e\in E\}
+% \end{align}
+% with
+% \begin{align}
+% a \cdot \bar{e} \cdot b = b^* \bar{e} a^* \;\;\;\; \forall a\in A, b \in
+% B
+% \end{align}
+%\end{definition}
+%$E^\circ$ is not a Hilbert bimodule for $(A, B)$ because it doesn't have a
+%natural $B$-valued inner product. But there is a $A$-valued inner product on
+%the left $A$-module $E^\circ$ with
+%\begin{align}
+% \langle \bar{e}_1, \bar{e}_2 \rangle = \langle e_2 , e_1 \rangle
+% \;\;\;\; e_1, e_2 \in E
+%\end{align}
+%and linearity in $A$:
+%\begin{align}
+% \langle a \bar{e}_1, \bar{e}_2 \rangle = a \langle \bar{e}_1, \bar{e}_2
+% \rangle \;\;\;\; \forall a \in A.
+%\end{align}
+%
+%%\begin{MyExercise}
+%% \textbf{Show that $E^\circ$ is a Hilbert bimodule $(B^{\circ}, A^{\circ})$
+%% }\newline
+%%
+%%
+%% Straightforward show properties of the Hilbert bimodule and its $B^{\circ}$
+%% valued inner product. Let $\bar{e}_1, \bar{e}_2 \in E^{\circ}$ and $a^\circ \in A,
+%% b^\circ \in B$. \\
+%% \begin{align}
+%% <\bar{e}_1, a^\circ \bar{e}_2> &= <\bar{e}_1, Ja^*J^{-1} \bar{e}_2>=\\
+%% &= <\bar{e}_1 , J a^* e_2> = \\
+%% &= <J^{-1} e_1, a^* e_2> =\\
+%% & = <a^* e_1, e_2>= <J^{-1}(a^\circ)^* J e_1, e_2> = \\
+%% & = <J^{-1} (a^\circ)^* \bar{e}_1, e_2> =\\
+%% & = <(a^\circ)^* \bar{e}_1 , \bar{e}_2>.
+%% \end{align}
+%%
+%% Next $<\bar{e}_1, \bar{e}_2 b^\circ> = <\bar{e}_1, \bar{e_2}> b^\circ$.
+%% \begin{align}
+%% <\bar{e}_1, \bar{e}_2 b^\circ> &= <\bar{e}_1, \bar{e}_2 Jb^*J^{-1}> =\\
+%% &= <\bar{e}_1, \bar{e_2}> Jb^*J^{-1} = \\
+%% &= <\bar{e}_1, \bar{e}_2> b^\circ.
+%% \end{align}
+%% Then:
+%% \begin{align}
+%% (<\bar{e}_1, \bar{e}_2)>_{E^\circ})^* &= (<e_2, e_1>_E)^* =\\
+%% &= <e_1, e_2>_E^* = <\bar{e}_2, \bar{e}_2>_{E^\circ}
+%% \end{align}
+%% And of course $<\bar{e}, \bar{e}> = <e, e> \geq 0$
+%%\end{MyExercise}
+%
+%\subsubsection{Construction of a Finite Real Spectral Triple from a Finite
+%Real Spectral Triple}
+%Given a Hilbert bimodule $E$ for $(B, A)$ we construct a spectral triple
+%$(B, H', D'; J', \gamma ')$ from $(A, H, D; J, \gamma)$
+%
+%For the $H'$ we make a $\mathbb{C}$-valued inner product on $H'$ by combining
+%the $A$ valued inner product on $E$ and $E^\circ$ with the
+%$\mathbb{C}$-valued inner product on $H$.
+%\begin{align}
+% H' := E\otimes _A H \otimes _A E^\circ
+%\end{align}
+%
+%Then the action of $B$ on $H'$ is:
+%\begin{align}
+% b(e_2 \otimes \xi \otimes \bar{e}_2 ) = (be_1) \otimes \xi \otimes
+% \bar{e}_2
+%\end{align}
+%The right action of $B$ on $H'$ defined by action on the right component
+%$E^\circ$
+%\begin{align}
+% J'(e_1 \otimes \xi \otimes \bar{e}_2) = e_2 \otimes J \xi \otimes
+% \bar{e}_1
+%\end{align}
+%with $b^\circ = J' b^* (J')^{-1}$, $b^* \in B$ action on $H'$.
+%\newline
+%
+%
+%\newpage
+%%\begin{MyExercise}
+%% \textbf{ Let $\nabla : E \Rightarrow E \otimes _A \Omega _d^1 (A)$ be a right connection on $E$
+%% consider the following anti-linear map:
+%% \begin{align}
+%% \tau : E \otimes_A \Omega _D^1 (A) &\rightarrow \Omega _D^1 (A) \otimes_A E^\circ\\
+%% e \otimes \omega &\mapsto -\omega ^* \otimes \bar{e}
+%% \end{align}
+%% Show that the map $\bar{\nabla} : E^\circ \rightarrow \Omega _D^1(A) \otimes E^\circ$
+%% with $\bar{\nabla}(\bar{e}) = \tau \circ \nabla(e)$ is a left connection, that means
+%% show that it satisfied the left Leibniz rule:
+%% \begin{equation}
+%% \bar{\nabla}(a\bar{e}) = [D, a] \otimes \bar{e} + a \bar{\nabla}(\bar{e})
+%% \end{equation}
+%% }
+%%
+%% Hagime:
+%% \begin{align}
+%% &\text{For one:}\\
+%% &\tau \circ \nabla(ae) = \bar{\nabla}(a\bar{e}) = \bar{\nabla}(a^* \bar{e})\\
+%% &\text{For two:}\\
+%% &\tau \circ \nabla(ae) = \tau(\nabla(e)a) + \tau \circ(e \otimes d(a))=\\
+%% &=a^*\bar{\nabla}(\bar{e}) - d(a)^* \otimes \bar{e}. \\
+%% &= a^*\bar{\nabla}(\bar{e}) + d(a^*) \otimes \bar{e}.
+%% \end{align}
+%%\end{MyExercise}
+%Then the connections
+%\begin{align}
+% &\nabla: E \rightarrow E\otimes _A \Omega _D ^1(A) \\
+% &\bar{\nabla}:E^\circ \rightarrow \Omega _D^1(A) \otimes _A E^\circ
+%\end{align}
+%give us the Dirac operator on $H' = E \otimes _A H \otimes _A E^\circ$
+%\begin{align}
+% D'(e_1 \otimes \xi \otimes \bar{e}_2) = (\nabla e_1) \xi \otimes
+% \bar{e_2}+ e_1 \otimes D\xi \otimes \bar{e}_2 + e_1 \otimes
+% \xi(\bar{\nabla}\bar{e}_2)
+%\end{align}
+%
+%And the right action of $\omega \in \Omega _D ^1(A)$ on $\xi \in H$ is
+%defined by
+%\begin{align}
+% \xi \mapsto \epsilon' J \omega ^* J^{-1}\xi
+%\end{align}
+%
+%Finally for the grading
+%\begin{align}
+% \gamma ' = 1 \otimes \gamma \otimes 1
+%\end{align}
+%
+%\begin{theorem}
+% Suppose $(A, H, D; J, \gamma)$ is a finite spectral triple of
+% $KO$-dimension $k$, let $\nabla$ be like above satisfying the
+% compatibility condition (like with finite spectral triples).
+%
+% Then $(B, H',D'; J', \gamma')$ is a finite spectral triple of
+% $KO$-Dimension $k$. ($H', D', J', \gamma'$ like above)
+%\end{theorem}
+%
+%\begin{proof}
+% The only thing left is to check if the $KO$-dimension is preserved,
+% for this we check if the $\epsilon$'s are the same.
+% \begin{align}
+% &(J')^2 = 1 \otimes J^2 \otimes 1 = \epsilon\\
+% &J' \gamma '= \epsilon ''\gamma'J'
+% \end{align}
+% and for $\epsilon '$
+% \begin{align}
+% J'D'(e_1 \otimes \xi \otimes \bar{e}_2)&=J'((\nabla e_1) \xi \otimes
+% \bar{e_2} + e_1 \otimes D\xi \otimes \bar{e}_2 + e_1 \otimes \xi (\tau
+% \nabla e_2))\\
+% &= \epsilon' D'(e_2 \otimes J\xi \otimes \bar{e}_2)\\
+% &= \epsilon' D'J'(e_1 \otimes \xi \bar{e}_2)
+% \end{align}
+%\end{proof}
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- <bcf:option type="singlevalued">
- <bcf:key>usenamea</bcf:key>
- <bcf:value>1</bcf:value>
- </bcf:option>
- <bcf:option type="singlevalued">
- <bcf:key>usenameb</bcf:key>
- <bcf:value>1</bcf:value>
- </bcf:option>
- <bcf:option type="singlevalued">
- <bcf:key>usenamec</bcf:key>
- <bcf:value>1</bcf:value>
- </bcf:option>
- <bcf:option type="singlevalued">
- <bcf:key>usetranslator</bcf:key>
- <bcf:value>0</bcf:value>
- </bcf:option>
- <bcf:option type="singlevalued">
- <bcf:key>useshortauthor</bcf:key>
- <bcf:value>1</bcf:value>
- </bcf:option>
- <bcf:option type="singlevalued">
- <bcf:key>useshorteditor</bcf:key>
- <bcf:value>1</bcf:value>
- </bcf:option>
- </bcf:options>
- <!-- BIBLATEX OPTION SCOPE -->
- <bcf:optionscope type="GLOBAL">
- <bcf:option datatype="xml">datamodel</bcf:option>
- <bcf:option datatype="xml">labelalphanametemplate</bcf:option>
- <bcf:option datatype="xml">labelalphatemplate</bcf:option>
- <bcf:option datatype="xml">inheritance</bcf:option>
- <bcf:option datatype="xml">translit</bcf:option>
- <bcf:option datatype="xml">uniquenametemplate</bcf:option>
- <bcf:option datatype="xml">sortingnamekeytemplate</bcf:option>
- <bcf:option datatype="xml">sortingtemplate</bcf:option>
- <bcf:option datatype="xml">extradatespec</bcf:option>
- <bcf:option datatype="xml">labelnamespec</bcf:option>
- <bcf:option datatype="xml">labeltitlespec</bcf:option>
- <bcf:option datatype="xml">labeldatespec</bcf:option>
- <bcf:option datatype="string">controlversion</bcf:option>
- <bcf:option datatype="string">alphaothers</bcf:option>
- <bcf:option datatype="string">sortalphaothers</bcf:option>
- <bcf:option datatype="string">presort</bcf:option>
- <bcf:option datatype="string">texencoding</bcf:option>
- <bcf:option datatype="string">bibencoding</bcf:option>
- <bcf:option datatype="string">sortingtemplatename</bcf:option>
- <bcf:option datatype="string">sortlocale</bcf:option>
- <bcf:option datatype="string">language</bcf:option>
- <bcf:option datatype="string">autolang</bcf:option>
- <bcf:option datatype="string">langhook</bcf:option>
- <bcf:option datatype="string">indexing</bcf:option>
- <bcf:option datatype="string">hyperref</bcf:option>
- <bcf:option datatype="string">backrefsetstyle</bcf:option>
- <bcf:option datatype="string">block</bcf:option>
- <bcf:option datatype="string">pagetracker</bcf:option>
- <bcf:option datatype="string">citecounter</bcf:option>
- <bcf:option datatype="string">citetracker</bcf:option>
- <bcf:option datatype="string">ibidtracker</bcf:option>
- <bcf:option datatype="string">idemtracker</bcf:option>
- <bcf:option datatype="string">opcittracker</bcf:option>
- <bcf:option datatype="string">loccittracker</bcf:option>
- <bcf:option datatype="string">labeldate</bcf:option>
- <bcf:option datatype="string">labeltime</bcf:option>
- <bcf:option datatype="string">dateera</bcf:option>
- <bcf:option datatype="string">date</bcf:option>
- <bcf:option datatype="string">time</bcf:option>
- <bcf:option datatype="string">eventdate</bcf:option>
- <bcf:option datatype="string">eventtime</bcf:option>
- <bcf:option datatype="string">origdate</bcf:option>
- <bcf:option datatype="string">origtime</bcf:option>
- <bcf:option datatype="string">urldate</bcf:option>
- <bcf:option datatype="string">urltime</bcf:option>
- <bcf:option datatype="string">alldatesusetime</bcf:option>
- <bcf:option datatype="string">alldates</bcf:option>
- <bcf:option datatype="string">alltimes</bcf:option>
- <bcf:option datatype="string">gregorianstart</bcf:option>
- <bcf:option datatype="string">autocite</bcf:option>
- <bcf:option datatype="string">notetype</bcf:option>
- <bcf:option datatype="string">uniquelist</bcf:option>
- <bcf:option datatype="string">uniquename</bcf:option>
- <bcf:option datatype="string">refsection</bcf:option>
- <bcf:option datatype="string">refsegment</bcf:option>
- <bcf:option datatype="string">citereset</bcf:option>
- <bcf:option datatype="string">sortlos</bcf:option>
- <bcf:option datatype="string">babel</bcf:option>
- <bcf:option datatype="string">datelabel</bcf:option>
- <bcf:option datatype="string">backrefstyle</bcf:option>
- <bcf:option datatype="string">arxiv</bcf:option>
- <bcf:option datatype="boolean">familyinits</bcf:option>
- <bcf:option datatype="boolean">giveninits</bcf:option>
- <bcf:option datatype="boolean">prefixinits</bcf:option>
- <bcf:option datatype="boolean">suffixinits</bcf:option>
- <bcf:option datatype="boolean">useafterword</bcf:option>
- <bcf:option datatype="boolean">useannotator</bcf:option>
- <bcf:option datatype="boolean">useauthor</bcf:option>
- <bcf:option datatype="boolean">usebookauthor</bcf:option>
- <bcf:option datatype="boolean">usecommentator</bcf:option>
- <bcf:option datatype="boolean">useeditor</bcf:option>
- <bcf:option datatype="boolean">useeditora</bcf:option>
- <bcf:option datatype="boolean">useeditorb</bcf:option>
- <bcf:option datatype="boolean">useeditorc</bcf:option>
- <bcf:option datatype="boolean">useforeword</bcf:option>
- <bcf:option datatype="boolean">useholder</bcf:option>
- <bcf:option datatype="boolean">useintroduction</bcf:option>
- <bcf:option datatype="boolean">usenamea</bcf:option>
- <bcf:option datatype="boolean">usenameb</bcf:option>
- <bcf:option datatype="boolean">usenamec</bcf:option>
- <bcf:option datatype="boolean">usetranslator</bcf:option>
- <bcf:option datatype="boolean">useshortauthor</bcf:option>
- <bcf:option datatype="boolean">useshorteditor</bcf:option>
- <bcf:option datatype="boolean">debug</bcf:option>
- <bcf:option datatype="boolean">loadfiles</bcf:option>
- <bcf:option datatype="boolean">safeinputenc</bcf:option>
- <bcf:option datatype="boolean">sortcase</bcf:option>
- <bcf:option datatype="boolean">sortupper</bcf:option>
- <bcf:option datatype="boolean">terseinits</bcf:option>
- <bcf:option datatype="boolean">abbreviate</bcf:option>
- <bcf:option datatype="boolean">dateabbrev</bcf:option>
- <bcf:option datatype="boolean">clearlang</bcf:option>
- <bcf:option datatype="boolean">sortcites</bcf:option>
- <bcf:option datatype="boolean">sortsets</bcf:option>
- <bcf:option datatype="boolean">backref</bcf:option>
- <bcf:option datatype="boolean">backreffloats</bcf:option>
- <bcf:option datatype="boolean">trackfloats</bcf:option>
- <bcf:option datatype="boolean">parentracker</bcf:option>
- <bcf:option datatype="boolean">labeldateusetime</bcf:option>
- <bcf:option datatype="boolean">datecirca</bcf:option>
- <bcf:option datatype="boolean">dateuncertain</bcf:option>
- <bcf:option datatype="boolean">dateusetime</bcf:option>
- <bcf:option datatype="boolean">eventdateusetime</bcf:option>
- <bcf:option datatype="boolean">origdateusetime</bcf:option>
- <bcf:option datatype="boolean">urldateusetime</bcf:option>
- <bcf:option datatype="boolean">julian</bcf:option>
- <bcf:option datatype="boolean">datezeros</bcf:option>
- <bcf:option datatype="boolean">timezeros</bcf:option>
- <bcf:option datatype="boolean">timezones</bcf:option>
- <bcf:option datatype="boolean">seconds</bcf:option>
- <bcf:option datatype="boolean">autopunct</bcf:option>
- <bcf:option datatype="boolean">punctfont</bcf:option>
- <bcf:option datatype="boolean">labelnumber</bcf:option>
- <bcf:option datatype="boolean">labelalpha</bcf:option>
- <bcf:option datatype="boolean">labeltitle</bcf:option>
- <bcf:option datatype="boolean">labeltitleyear</bcf:option>
- <bcf:option datatype="boolean">labeldateparts</bcf:option>
- <bcf:option datatype="boolean">nohashothers</bcf:option>
- <bcf:option datatype="boolean">nosortothers</bcf:option>
- <bcf:option datatype="boolean">noroman</bcf:option>
- <bcf:option datatype="boolean">singletitle</bcf:option>
- <bcf:option datatype="boolean">uniquetitle</bcf:option>
- <bcf:option datatype="boolean">uniquebaretitle</bcf:option>
- <bcf:option datatype="boolean">uniquework</bcf:option>
- <bcf:option datatype="boolean">uniqueprimaryauthor</bcf:option>
- <bcf:option datatype="boolean">defernumbers</bcf:option>
- <bcf:option datatype="boolean">locallabelwidth</bcf:option>
- <bcf:option datatype="boolean">bibwarn</bcf:option>
- <bcf:option datatype="boolean">useprefix</bcf:option>
- <bcf:option datatype="boolean">skipbib</bcf:option>
- <bcf:option datatype="boolean">skipbiblist</bcf:option>
- <bcf:option datatype="boolean">skiplab</bcf:option>
- <bcf:option datatype="boolean">dataonly</bcf:option>
- <bcf:option datatype="boolean">defernums</bcf:option>
- <bcf:option datatype="boolean">firstinits</bcf:option>
- <bcf:option datatype="boolean">sortfirstinits</bcf:option>
- <bcf:option datatype="boolean">sortgiveninits</bcf:option>
- <bcf:option datatype="boolean">labelyear</bcf:option>
- <bcf:option datatype="boolean">isbn</bcf:option>
- <bcf:option datatype="boolean">url</bcf:option>
- <bcf:option datatype="boolean">doi</bcf:option>
- <bcf:option datatype="boolean">eprint</bcf:option>
- <bcf:option datatype="boolean">related</bcf:option>
- <bcf:option datatype="boolean">subentry</bcf:option>
- <bcf:option datatype="boolean">bibtexcaseprotection</bcf:option>
- <bcf:option datatype="integer">mincrossrefs</bcf:option>
- <bcf:option datatype="integer">minxrefs</bcf:option>
- <bcf:option datatype="integer">maxnames</bcf:option>
- <bcf:option datatype="integer">minnames</bcf:option>
- <bcf:option datatype="integer">maxbibnames</bcf:option>
- <bcf:option datatype="integer">minbibnames</bcf:option>
- <bcf:option datatype="integer">maxcitenames</bcf:option>
- <bcf:option datatype="integer">mincitenames</bcf:option>
- <bcf:option datatype="integer">maxsortnames</bcf:option>
- <bcf:option datatype="integer">minsortnames</bcf:option>
- <bcf:option datatype="integer">maxitems</bcf:option>
- <bcf:option datatype="integer">minitems</bcf:option>
- <bcf:option datatype="integer">maxalphanames</bcf:option>
- <bcf:option datatype="integer">minalphanames</bcf:option>
- <bcf:option datatype="integer">maxparens</bcf:option>
- <bcf:option datatype="integer">dateeraauto</bcf:option>
- </bcf:optionscope>
- <bcf:optionscope type="ENTRYTYPE">
- <bcf:option datatype="string">alphaothers</bcf:option>
- <bcf:option datatype="string">sortalphaothers</bcf:option>
- <bcf:option datatype="string">presort</bcf:option>
- <bcf:option datatype="string">indexing</bcf:option>
- <bcf:option datatype="string">citetracker</bcf:option>
- <bcf:option datatype="string">ibidtracker</bcf:option>
- <bcf:option datatype="string">idemtracker</bcf:option>
- <bcf:option datatype="string">opcittracker</bcf:option>
- <bcf:option datatype="string">loccittracker</bcf:option>
- <bcf:option datatype="string">uniquelist</bcf:option>
- <bcf:option datatype="string">uniquename</bcf:option>
- <bcf:option datatype="boolean">familyinits</bcf:option>
- <bcf:option datatype="boolean">giveninits</bcf:option>
- <bcf:option datatype="boolean">prefixinits</bcf:option>
- <bcf:option datatype="boolean">suffixinits</bcf:option>
- <bcf:option datatype="boolean">useafterword</bcf:option>
- <bcf:option datatype="boolean">useannotator</bcf:option>
- <bcf:option datatype="boolean">useauthor</bcf:option>
- <bcf:option datatype="boolean">usebookauthor</bcf:option>
- <bcf:option datatype="boolean">usecommentator</bcf:option>
- <bcf:option datatype="boolean">useeditor</bcf:option>
- <bcf:option datatype="boolean">useeditora</bcf:option>
- <bcf:option datatype="boolean">useeditorb</bcf:option>
- <bcf:option datatype="boolean">useeditorc</bcf:option>
- <bcf:option datatype="boolean">useforeword</bcf:option>
- <bcf:option datatype="boolean">useholder</bcf:option>
- <bcf:option datatype="boolean">useintroduction</bcf:option>
- <bcf:option datatype="boolean">usenamea</bcf:option>
- <bcf:option datatype="boolean">usenameb</bcf:option>
- <bcf:option datatype="boolean">usenamec</bcf:option>
- <bcf:option datatype="boolean">usetranslator</bcf:option>
- <bcf:option datatype="boolean">useshortauthor</bcf:option>
- <bcf:option datatype="boolean">useshorteditor</bcf:option>
- <bcf:option datatype="boolean">terseinits</bcf:option>
- <bcf:option datatype="boolean">abbreviate</bcf:option>
- <bcf:option datatype="boolean">dateabbrev</bcf:option>
- <bcf:option datatype="boolean">clearlang</bcf:option>
- <bcf:option datatype="boolean">labelnumber</bcf:option>
- <bcf:option datatype="boolean">labelalpha</bcf:option>
- <bcf:option datatype="boolean">labeltitle</bcf:option>
- <bcf:option datatype="boolean">labeltitleyear</bcf:option>
- <bcf:option datatype="boolean">labeldateparts</bcf:option>
- <bcf:option datatype="boolean">nohashothers</bcf:option>
- <bcf:option datatype="boolean">nosortothers</bcf:option>
- <bcf:option datatype="boolean">noroman</bcf:option>
- <bcf:option datatype="boolean">singletitle</bcf:option>
- <bcf:option datatype="boolean">uniquetitle</bcf:option>
- <bcf:option datatype="boolean">uniquebaretitle</bcf:option>
- <bcf:option datatype="boolean">uniquework</bcf:option>
- <bcf:option datatype="boolean">uniqueprimaryauthor</bcf:option>
- <bcf:option datatype="boolean">useprefix</bcf:option>
- <bcf:option datatype="boolean">skipbib</bcf:option>
- <bcf:option datatype="boolean">skipbiblist</bcf:option>
- <bcf:option datatype="boolean">skiplab</bcf:option>
- <bcf:option datatype="boolean">dataonly</bcf:option>
- <bcf:option datatype="boolean">skiplos</bcf:option>
- <bcf:option datatype="boolean">labelyear</bcf:option>
- <bcf:option datatype="boolean">isbn</bcf:option>
- <bcf:option datatype="boolean">url</bcf:option>
- <bcf:option datatype="boolean">doi</bcf:option>
- <bcf:option datatype="boolean">eprint</bcf:option>
- <bcf:option datatype="boolean">related</bcf:option>
- <bcf:option datatype="boolean">subentry</bcf:option>
- <bcf:option datatype="boolean">bibtexcaseprotection</bcf:option>
- <bcf:option datatype="xml">labelalphatemplate</bcf:option>
- <bcf:option datatype="xml">translit</bcf:option>
- <bcf:option datatype="xml">sortexclusion</bcf:option>
- <bcf:option datatype="xml">sortinclusion</bcf:option>
- <bcf:option datatype="xml">labelnamespec</bcf:option>
- <bcf:option datatype="xml">labeltitlespec</bcf:option>
- <bcf:option datatype="xml">labeldatespec</bcf:option>
- <bcf:option datatype="integer">maxnames</bcf:option>
- <bcf:option datatype="integer">minnames</bcf:option>
- <bcf:option datatype="integer">maxbibnames</bcf:option>
- <bcf:option datatype="integer">minbibnames</bcf:option>
- <bcf:option datatype="integer">maxcitenames</bcf:option>
- <bcf:option datatype="integer">mincitenames</bcf:option>
- <bcf:option datatype="integer">maxsortnames</bcf:option>
- <bcf:option datatype="integer">minsortnames</bcf:option>
- <bcf:option datatype="integer">maxitems</bcf:option>
- <bcf:option datatype="integer">minitems</bcf:option>
- <bcf:option datatype="integer">maxalphanames</bcf:option>
- <bcf:option datatype="integer">minalphanames</bcf:option>
- </bcf:optionscope>
- <bcf:optionscope type="ENTRY">
- <bcf:option datatype="string">noinherit</bcf:option>
- <bcf:option datatype="string" backendin="sortingnamekeytemplatename,uniquenametemplatename,labelalphanametemplatename">nametemplates</bcf:option>
- <bcf:option datatype="string" backendout="1">labelalphanametemplatename</bcf:option>
- <bcf:option datatype="string" backendout="1">uniquenametemplatename</bcf:option>
- <bcf:option datatype="string" backendout="1">sortingnamekeytemplatename</bcf:option>
- <bcf:option datatype="string">presort</bcf:option>
- <bcf:option datatype="string" backendout="1">indexing</bcf:option>
- <bcf:option datatype="string" backendout="1">citetracker</bcf:option>
- <bcf:option datatype="string" backendout="1">ibidtracker</bcf:option>
- <bcf:option datatype="string" backendout="1">idemtracker</bcf:option>
- <bcf:option datatype="string" backendout="1">opcittracker</bcf:option>
- <bcf:option datatype="string" backendout="1">loccittracker</bcf:option>
- <bcf:option datatype="string">uniquelist</bcf:option>
- <bcf:option datatype="string">uniquename</bcf:option>
- <bcf:option datatype="boolean" backendout="1">familyinits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">giveninits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">prefixinits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">suffixinits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useafterword</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useannotator</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useauthor</bcf:option>
- <bcf:option datatype="boolean" backendout="1">usebookauthor</bcf:option>
- <bcf:option datatype="boolean" backendout="1">usecommentator</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useeditor</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useeditora</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useeditorb</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useeditorc</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useforeword</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useholder</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useintroduction</bcf:option>
- <bcf:option datatype="boolean" backendout="1">usenamea</bcf:option>
- <bcf:option datatype="boolean" backendout="1">usenameb</bcf:option>
- <bcf:option datatype="boolean" backendout="1">usenamec</bcf:option>
- <bcf:option datatype="boolean" backendout="1">usetranslator</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useshortauthor</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useshorteditor</bcf:option>
- <bcf:option datatype="boolean" backendout="1">terseinits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">abbreviate</bcf:option>
- <bcf:option datatype="boolean" backendout="1">dateabbrev</bcf:option>
- <bcf:option datatype="boolean" backendout="1">clearlang</bcf:option>
- <bcf:option datatype="boolean" backendout="1">labelnumber</bcf:option>
- <bcf:option datatype="boolean" backendout="1">labelalpha</bcf:option>
- <bcf:option datatype="boolean" backendout="1">labeltitle</bcf:option>
- <bcf:option datatype="boolean" backendout="1">labeltitleyear</bcf:option>
- <bcf:option datatype="boolean" backendout="1">labeldateparts</bcf:option>
- <bcf:option datatype="boolean">nohashothers</bcf:option>
- <bcf:option datatype="boolean">nosortothers</bcf:option>
- <bcf:option datatype="boolean">noroman</bcf:option>
- <bcf:option datatype="boolean">singletitle</bcf:option>
- <bcf:option datatype="boolean">uniquetitle</bcf:option>
- <bcf:option datatype="boolean">uniquebaretitle</bcf:option>
- <bcf:option datatype="boolean">uniquework</bcf:option>
- <bcf:option datatype="boolean">uniqueprimaryauthor</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useprefix</bcf:option>
- <bcf:option datatype="boolean" backendout="1">skipbib</bcf:option>
- <bcf:option datatype="boolean" backendout="1">skipbiblist</bcf:option>
- <bcf:option datatype="boolean" backendout="1">skiplab</bcf:option>
- <bcf:option datatype="boolean" backendin="uniquename=false,uniquelist=false,skipbib=true,skipbiblist=true,skiplab=true">dataonly</bcf:option>
- <bcf:option datatype="boolean" backendout="1">skiplos</bcf:option>
- <bcf:option datatype="boolean" backendout="1">isbn</bcf:option>
- <bcf:option datatype="boolean" backendout="1">url</bcf:option>
- <bcf:option datatype="boolean" backendout="1">doi</bcf:option>
- <bcf:option datatype="boolean" backendout="1">eprint</bcf:option>
- <bcf:option datatype="boolean" backendout="1">related</bcf:option>
- <bcf:option datatype="boolean" backendout="1">subentry</bcf:option>
- <bcf:option datatype="boolean" backendout="1">bibtexcaseprotection</bcf:option>
- <bcf:option datatype="integer" backendin="maxcitenames,maxbibnames,maxsortnames">maxnames</bcf:option>
- <bcf:option datatype="integer" backendin="mincitenames,minbibnames,minsortnames">minnames</bcf:option>
- <bcf:option datatype="integer" backendout="1">maxbibnames</bcf:option>
- <bcf:option datatype="integer" backendout="1">minbibnames</bcf:option>
- <bcf:option datatype="integer" backendout="1">maxcitenames</bcf:option>
- <bcf:option datatype="integer" backendout="1">mincitenames</bcf:option>
- <bcf:option datatype="integer" backendout="1">maxsortnames</bcf:option>
- <bcf:option datatype="integer" backendout="1">minsortnames</bcf:option>
- <bcf:option datatype="integer" backendout="1">maxitems</bcf:option>
- <bcf:option datatype="integer" backendout="1">minitems</bcf:option>
- <bcf:option datatype="integer" backendout="1">maxalphanames</bcf:option>
- <bcf:option datatype="integer" backendout="1">minalphanames</bcf:option>
- </bcf:optionscope>
- <bcf:optionscope type="NAMELIST">
- <bcf:option datatype="string" backendin="sortingnamekeytemplatename,uniquenametemplatename,labelalphanametemplatename">nametemplates</bcf:option>
- <bcf:option datatype="string" backendout="1">labelalphanametemplatename</bcf:option>
- <bcf:option datatype="string" backendout="1">uniquenametemplatename</bcf:option>
- <bcf:option datatype="string" backendout="1">sortingnamekeytemplatename</bcf:option>
- <bcf:option datatype="string">uniquelist</bcf:option>
- <bcf:option datatype="string">uniquename</bcf:option>
- <bcf:option datatype="boolean" backendout="1">familyinits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">giveninits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">prefixinits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">suffixinits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">terseinits</bcf:option>
- <bcf:option datatype="boolean">nohashothers</bcf:option>
- <bcf:option datatype="boolean">nosortothers</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useprefix</bcf:option>
- </bcf:optionscope>
- <bcf:optionscope type="NAME">
- <bcf:option datatype="string" backendin="sortingnamekeytemplatename,uniquenametemplatename,labelalphanametemplatename">nametemplates</bcf:option>
- <bcf:option datatype="string" backendout="1">labelalphanametemplatename</bcf:option>
- <bcf:option datatype="string" backendout="1">uniquenametemplatename</bcf:option>
- <bcf:option datatype="string" backendout="1">sortingnamekeytemplatename</bcf:option>
- <bcf:option datatype="string">uniquename</bcf:option>
- <bcf:option datatype="boolean" backendout="1">familyinits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">giveninits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">prefixinits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">suffixinits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">terseinits</bcf:option>
- <bcf:option datatype="boolean" backendout="1">useprefix</bcf:option>
- </bcf:optionscope>
- <!-- DATAFIELDSETS -->
- <bcf:datafieldset name="setnames">
- <bcf:member datatype="name" fieldtype="list"/>
- </bcf:datafieldset>
- <bcf:datafieldset name="settitles">
- <bcf:member field="title"/>
- <bcf:member field="booktitle"/>
- <bcf:member field="eventtitle"/>
- <bcf:member field="issuetitle"/>
- <bcf:member field="journaltitle"/>
- <bcf:member field="maintitle"/>
- <bcf:member field="origtitle"/>
- </bcf:datafieldset>
- <!-- SOURCEMAP -->
- <bcf:sourcemap>
- <bcf:maps datatype="bibtex" level="driver">
- <bcf:map>
- <bcf:map_step map_field_set="day" map_null="1"/>
- </bcf:map>
- <bcf:map>
- <bcf:map_step map_type_source="conference" map_type_target="inproceedings"/>
- <bcf:map_step map_type_source="electronic" map_type_target="online"/>
- <bcf:map_step map_type_source="www" map_type_target="online"/>
- </bcf:map>
- <bcf:map>
- <bcf:map_step map_type_source="mastersthesis" map_type_target="thesis" map_final="1"/>
- <bcf:map_step map_field_set="type" map_field_value="mathesis"/>
- </bcf:map>
- <bcf:map>
- <bcf:map_step map_type_source="phdthesis" map_type_target="thesis" map_final="1"/>
- <bcf:map_step map_field_set="type" map_field_value="phdthesis"/>
- </bcf:map>
- <bcf:map>
- <bcf:map_step map_type_source="techreport" map_type_target="report" map_final="1"/>
- <bcf:map_step map_field_set="type" map_field_value="techreport"/>
- </bcf:map>
- <bcf:map>
- <bcf:map_step map_field_source="hyphenation" map_field_target="langid"/>
- <bcf:map_step map_field_source="address" map_field_target="location"/>
- <bcf:map_step map_field_source="school" map_field_target="institution"/>
- <bcf:map_step map_field_source="annote" map_field_target="annotation"/>
- <bcf:map_step map_field_source="archiveprefix" map_field_target="eprinttype"/>
- <bcf:map_step map_field_source="journal" map_field_target="journaltitle"/>
- <bcf:map_step map_field_source="primaryclass" map_field_target="eprintclass"/>
- <bcf:map_step map_field_source="key" map_field_target="sortkey"/>
- <bcf:map_step map_field_source="pdf" map_field_target="file"/>
- </bcf:map>
- </bcf:maps>
- </bcf:sourcemap>
- <!-- LABELALPHA NAME TEMPLATE -->
- <bcf:labelalphanametemplate name="global">
- <bcf:namepart order="1" use="1" pre="1" substring_width="1" substring_compound="1">prefix</bcf:namepart>
- <bcf:namepart order="2">family</bcf:namepart>
- </bcf:labelalphanametemplate>
- <!-- LABELALPHA TEMPLATE -->
- <bcf:labelalphatemplate type="global">
- <bcf:labelelement order="1">
- <bcf:labelpart final="1">shorthand</bcf:labelpart>
- <bcf:labelpart>label</bcf:labelpart>
- <bcf:labelpart substring_width="3" substring_side="left" ifnames="1">labelname</bcf:labelpart>
- <bcf:labelpart substring_width="1" substring_side="left">labelname</bcf:labelpart>
- </bcf:labelelement>
- <bcf:labelelement order="2">
- <bcf:labelpart substring_width="2" substring_side="right">year</bcf:labelpart>
- </bcf:labelelement>
- </bcf:labelalphatemplate>
- <!-- EXTRADATE -->
- <bcf:extradatespec>
- <bcf:scope>
- <bcf:field order="1">labelyear</bcf:field>
- <bcf:field order="2">year</bcf:field>
- </bcf:scope>
- </bcf:extradatespec>
- <!-- INHERITANCE -->
- <bcf:inheritance>
- <bcf:defaults inherit_all="true" override_target="false">
- </bcf:defaults>
- <bcf:inherit>
- <bcf:type_pair source="mvbook" target="inbook"/>
- <bcf:type_pair source="mvbook" target="bookinbook"/>
- <bcf:type_pair source="mvbook" target="suppbook"/>
- <bcf:type_pair source="book" target="inbook"/>
- <bcf:type_pair source="book" target="bookinbook"/>
- <bcf:type_pair source="book" target="suppbook"/>
- <bcf:field source="author" target="author"/>
- <bcf:field source="author" target="bookauthor"/>
- </bcf:inherit>
- <bcf:inherit>
- <bcf:type_pair source="mvbook" target="book"/>
- <bcf:type_pair source="mvbook" target="inbook"/>
- <bcf:type_pair source="mvbook" target="bookinbook"/>
- <bcf:type_pair source="mvbook" target="suppbook"/>
- <bcf:field source="title" target="maintitle"/>
- <bcf:field source="subtitle" target="mainsubtitle"/>
- <bcf:field source="titleaddon" target="maintitleaddon"/>
- <bcf:field source="shorttitle" skip="true"/>
- <bcf:field source="sorttitle" skip="true"/>
- <bcf:field source="indextitle" skip="true"/>
- <bcf:field source="indexsorttitle" skip="true"/>
- </bcf:inherit>
- <bcf:inherit>
- <bcf:type_pair source="mvcollection" target="collection"/>
- <bcf:type_pair source="mvcollection" target="reference"/>
- <bcf:type_pair source="mvcollection" target="incollection"/>
- <bcf:type_pair source="mvcollection" target="inreference"/>
- <bcf:type_pair source="mvcollection" target="suppcollection"/>
- <bcf:type_pair source="mvreference" target="collection"/>
- <bcf:type_pair source="mvreference" target="reference"/>
- <bcf:type_pair source="mvreference" target="incollection"/>
- <bcf:type_pair source="mvreference" target="inreference"/>
- <bcf:type_pair source="mvreference" target="suppcollection"/>
- <bcf:field source="title" target="maintitle"/>
- <bcf:field source="subtitle" target="mainsubtitle"/>
- <bcf:field source="titleaddon" target="maintitleaddon"/>
- <bcf:field source="shorttitle" skip="true"/>
- <bcf:field source="sorttitle" skip="true"/>
- <bcf:field source="indextitle" skip="true"/>
- <bcf:field source="indexsorttitle" skip="true"/>
- </bcf:inherit>
- <bcf:inherit>
- <bcf:type_pair source="mvproceedings" target="proceedings"/>
- <bcf:type_pair source="mvproceedings" target="inproceedings"/>
- <bcf:field source="title" target="maintitle"/>
- <bcf:field source="subtitle" target="mainsubtitle"/>
- <bcf:field source="titleaddon" target="maintitleaddon"/>
- <bcf:field source="shorttitle" skip="true"/>
- <bcf:field source="sorttitle" skip="true"/>
- <bcf:field source="indextitle" skip="true"/>
- <bcf:field source="indexsorttitle" skip="true"/>
- </bcf:inherit>
- <bcf:inherit>
- <bcf:type_pair source="book" target="inbook"/>
- <bcf:type_pair source="book" target="bookinbook"/>
- <bcf:type_pair source="book" target="suppbook"/>
- <bcf:field source="title" target="booktitle"/>
- <bcf:field source="subtitle" target="booksubtitle"/>
- <bcf:field source="titleaddon" target="booktitleaddon"/>
- <bcf:field source="shorttitle" skip="true"/>
- <bcf:field source="sorttitle" skip="true"/>
- <bcf:field source="indextitle" skip="true"/>
- <bcf:field source="indexsorttitle" skip="true"/>
- </bcf:inherit>
- <bcf:inherit>
- <bcf:type_pair source="collection" target="incollection"/>
- <bcf:type_pair source="collection" target="inreference"/>
- <bcf:type_pair source="collection" target="suppcollection"/>
- <bcf:type_pair source="reference" target="incollection"/>
- <bcf:type_pair source="reference" target="inreference"/>
- <bcf:type_pair source="reference" target="suppcollection"/>
- <bcf:field source="title" target="booktitle"/>
- <bcf:field source="subtitle" target="booksubtitle"/>
- <bcf:field source="titleaddon" target="booktitleaddon"/>
- <bcf:field source="shorttitle" skip="true"/>
- <bcf:field source="sorttitle" skip="true"/>
- <bcf:field source="indextitle" skip="true"/>
- <bcf:field source="indexsorttitle" skip="true"/>
- </bcf:inherit>
- <bcf:inherit>
- <bcf:type_pair source="proceedings" target="inproceedings"/>
- <bcf:field source="title" target="booktitle"/>
- <bcf:field source="subtitle" target="booksubtitle"/>
- <bcf:field source="titleaddon" target="booktitleaddon"/>
- <bcf:field source="shorttitle" skip="true"/>
- <bcf:field source="sorttitle" skip="true"/>
- <bcf:field source="indextitle" skip="true"/>
- <bcf:field source="indexsorttitle" skip="true"/>
- </bcf:inherit>
- <bcf:inherit>
- <bcf:type_pair source="periodical" target="article"/>
- <bcf:type_pair source="periodical" target="suppperiodical"/>
- <bcf:field source="title" target="journaltitle"/>
- <bcf:field source="subtitle" target="journalsubtitle"/>
- <bcf:field source="titleaddon" target="journaltitleaddon"/>
- <bcf:field source="shorttitle" skip="true"/>
- <bcf:field source="sorttitle" skip="true"/>
- <bcf:field source="indextitle" skip="true"/>
- <bcf:field source="indexsorttitle" skip="true"/>
- </bcf:inherit>
- <bcf:inherit>
- <bcf:type_pair source="*" target="*"/>
- <bcf:field source="ids" skip="true"/>
- <bcf:field source="crossref" skip="true"/>
- <bcf:field source="xref" skip="true"/>
- <bcf:field source="entryset" skip="true"/>
- <bcf:field source="entrysubtype" skip="true"/>
- <bcf:field source="execute" skip="true"/>
- <bcf:field source="label" skip="true"/>
- <bcf:field source="options" skip="true"/>
- <bcf:field source="presort" skip="true"/>
- <bcf:field source="related" skip="true"/>
- <bcf:field source="relatedoptions" skip="true"/>
- <bcf:field source="relatedstring" skip="true"/>
- <bcf:field source="relatedtype" skip="true"/>
- <bcf:field source="shorthand" skip="true"/>
- <bcf:field source="shorthandintro" skip="true"/>
- <bcf:field source="sortkey" skip="true"/>
- </bcf:inherit>
- </bcf:inheritance>
- <!-- UNIQUENAME TEMPLATES -->
- <bcf:uniquenametemplate name="global">
- <bcf:namepart order="1" use="1" base="1">prefix</bcf:namepart>
- <bcf:namepart order="2" base="1">family</bcf:namepart>
- <bcf:namepart order="3">given</bcf:namepart>
- </bcf:uniquenametemplate>
- <!-- SORTING NAME KEY TEMPLATES -->
- <bcf:sortingnamekeytemplate name="global">
- <bcf:keypart order="1">
- <bcf:part type="namepart" order="1" use="1">prefix</bcf:part>
- <bcf:part type="namepart" order="2">family</bcf:part>
- </bcf:keypart>
- <bcf:keypart order="2">
- <bcf:part type="namepart" order="1">given</bcf:part>
- </bcf:keypart>
- <bcf:keypart order="3">
- <bcf:part type="namepart" order="1">suffix</bcf:part>
- </bcf:keypart>
- <bcf:keypart order="4">
- <bcf:part type="namepart" order="1" use="0">prefix</bcf:part>
- </bcf:keypart>
- </bcf:sortingnamekeytemplate>
- <bcf:presort>mm</bcf:presort>
- <!-- DATA MODEL -->
- <bcf:datamodel>
- <bcf:constants>
- <bcf:constant type="list" name="gender">sf,sm,sn,pf,pm,pn,pp</bcf:constant>
- <bcf:constant type="list" name="nameparts">family,given,prefix,suffix</bcf:constant>
- <bcf:constant type="list" name="optiondatatypes">boolean,integer,string,xml</bcf:constant>
- <bcf:constant type="list" name="multiscriptforms">default,transliteration,transcription,translation</bcf:constant>
- </bcf:constants>
- <bcf:entrytypes>
- <bcf:entrytype>article</bcf:entrytype>
- <bcf:entrytype>artwork</bcf:entrytype>
- <bcf:entrytype>audio</bcf:entrytype>
- <bcf:entrytype>bibnote</bcf:entrytype>
- <bcf:entrytype>book</bcf:entrytype>
- <bcf:entrytype>bookinbook</bcf:entrytype>
- <bcf:entrytype>booklet</bcf:entrytype>
- <bcf:entrytype>collection</bcf:entrytype>
- <bcf:entrytype>commentary</bcf:entrytype>
- <bcf:entrytype>customa</bcf:entrytype>
- <bcf:entrytype>customb</bcf:entrytype>
- <bcf:entrytype>customc</bcf:entrytype>
- <bcf:entrytype>customd</bcf:entrytype>
- <bcf:entrytype>custome</bcf:entrytype>
- <bcf:entrytype>customf</bcf:entrytype>
- <bcf:entrytype>dataset</bcf:entrytype>
- <bcf:entrytype>inbook</bcf:entrytype>
- <bcf:entrytype>incollection</bcf:entrytype>
- <bcf:entrytype>inproceedings</bcf:entrytype>
- <bcf:entrytype>inreference</bcf:entrytype>
- <bcf:entrytype>image</bcf:entrytype>
- <bcf:entrytype>jurisdiction</bcf:entrytype>
- <bcf:entrytype>legal</bcf:entrytype>
- <bcf:entrytype>legislation</bcf:entrytype>
- <bcf:entrytype>letter</bcf:entrytype>
- <bcf:entrytype>manual</bcf:entrytype>
- <bcf:entrytype>misc</bcf:entrytype>
- <bcf:entrytype>movie</bcf:entrytype>
- <bcf:entrytype>music</bcf:entrytype>
- <bcf:entrytype>mvcollection</bcf:entrytype>
- <bcf:entrytype>mvreference</bcf:entrytype>
- <bcf:entrytype>mvproceedings</bcf:entrytype>
- <bcf:entrytype>mvbook</bcf:entrytype>
- <bcf:entrytype>online</bcf:entrytype>
- <bcf:entrytype>patent</bcf:entrytype>
- <bcf:entrytype>performance</bcf:entrytype>
- <bcf:entrytype>periodical</bcf:entrytype>
- <bcf:entrytype>proceedings</bcf:entrytype>
- <bcf:entrytype>reference</bcf:entrytype>
- <bcf:entrytype>report</bcf:entrytype>
- <bcf:entrytype>review</bcf:entrytype>
- <bcf:entrytype>set</bcf:entrytype>
- <bcf:entrytype>software</bcf:entrytype>
- <bcf:entrytype>standard</bcf:entrytype>
- <bcf:entrytype>suppbook</bcf:entrytype>
- <bcf:entrytype>suppcollection</bcf:entrytype>
- <bcf:entrytype>suppperiodical</bcf:entrytype>
- <bcf:entrytype>thesis</bcf:entrytype>
- <bcf:entrytype>unpublished</bcf:entrytype>
- <bcf:entrytype>video</bcf:entrytype>
- <bcf:entrytype skip_output="true">xdata</bcf:entrytype>
- </bcf:entrytypes>
- <bcf:fields>
- <bcf:field fieldtype="field" datatype="integer">sortyear</bcf:field>
- <bcf:field fieldtype="field" datatype="integer">volume</bcf:field>
- <bcf:field fieldtype="field" datatype="integer">volumes</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">abstract</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">addendum</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">annotation</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">booksubtitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">booktitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">booktitleaddon</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">chapter</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">edition</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">eid</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">entrysubtype</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">eprintclass</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">eprinttype</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">eventtitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">eventtitleaddon</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">gender</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">howpublished</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">indexsorttitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">indextitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">isan</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">isbn</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">ismn</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">isrn</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">issn</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">issue</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">issuesubtitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">issuetitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">issuetitleaddon</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">iswc</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">journalsubtitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">journaltitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">journaltitleaddon</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">label</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">langid</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">langidopts</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">library</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">mainsubtitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">maintitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">maintitleaddon</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">nameaddon</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">note</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">number</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">origtitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">pagetotal</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">part</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">relatedstring</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">relatedtype</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">reprinttitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">series</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">shorthandintro</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">subtitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">title</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">titleaddon</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">usera</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">userb</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">userc</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">userd</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">usere</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">userf</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">venue</bcf:field>
- <bcf:field fieldtype="field" datatype="literal">version</bcf:field>
- <bcf:field fieldtype="field" datatype="literal" label="true">shorthand</bcf:field>
- <bcf:field fieldtype="field" datatype="literal" label="true">shortjournal</bcf:field>
- <bcf:field fieldtype="field" datatype="literal" label="true">shortseries</bcf:field>
- <bcf:field fieldtype="field" datatype="literal" label="true">shorttitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal" skip_output="true">sorttitle</bcf:field>
- <bcf:field fieldtype="field" datatype="literal" skip_output="true">sortshorthand</bcf:field>
- <bcf:field fieldtype="field" datatype="literal" skip_output="true">sortkey</bcf:field>
- <bcf:field fieldtype="field" datatype="literal" skip_output="true">presort</bcf:field>
- <bcf:field fieldtype="list" datatype="literal">institution</bcf:field>
- <bcf:field fieldtype="list" datatype="literal">lista</bcf:field>
- <bcf:field fieldtype="list" datatype="literal">listb</bcf:field>
- <bcf:field fieldtype="list" datatype="literal">listc</bcf:field>
- <bcf:field fieldtype="list" datatype="literal">listd</bcf:field>
- <bcf:field fieldtype="list" datatype="literal">liste</bcf:field>
- <bcf:field fieldtype="list" datatype="literal">listf</bcf:field>
- <bcf:field fieldtype="list" datatype="literal">location</bcf:field>
- <bcf:field fieldtype="list" datatype="literal">organization</bcf:field>
- <bcf:field fieldtype="list" datatype="literal">origlocation</bcf:field>
- <bcf:field fieldtype="list" datatype="literal">origpublisher</bcf:field>
- <bcf:field fieldtype="list" datatype="literal">publisher</bcf:field>
- <bcf:field fieldtype="list" datatype="name">afterword</bcf:field>
- <bcf:field fieldtype="list" datatype="name">annotator</bcf:field>
- <bcf:field fieldtype="list" datatype="name">author</bcf:field>
- <bcf:field fieldtype="list" datatype="name">bookauthor</bcf:field>
- <bcf:field fieldtype="list" datatype="name">commentator</bcf:field>
- <bcf:field fieldtype="list" datatype="name">editor</bcf:field>
- <bcf:field fieldtype="list" datatype="name">editora</bcf:field>
- <bcf:field fieldtype="list" datatype="name">editorb</bcf:field>
- <bcf:field fieldtype="list" datatype="name">editorc</bcf:field>
- <bcf:field fieldtype="list" datatype="name">foreword</bcf:field>
- <bcf:field fieldtype="list" datatype="name">holder</bcf:field>
- <bcf:field fieldtype="list" datatype="name">introduction</bcf:field>
- <bcf:field fieldtype="list" datatype="name">namea</bcf:field>
- <bcf:field fieldtype="list" datatype="name">nameb</bcf:field>
- <bcf:field fieldtype="list" datatype="name">namec</bcf:field>
- <bcf:field fieldtype="list" datatype="name">translator</bcf:field>
- <bcf:field fieldtype="list" datatype="name" label="true">shortauthor</bcf:field>
- <bcf:field fieldtype="list" datatype="name" label="true">shorteditor</bcf:field>
- <bcf:field fieldtype="list" datatype="name" skip_output="true">sortname</bcf:field>
- <bcf:field fieldtype="field" datatype="key">authortype</bcf:field>
- <bcf:field fieldtype="field" datatype="key">editoratype</bcf:field>
- <bcf:field fieldtype="field" datatype="key">editorbtype</bcf:field>
- <bcf:field fieldtype="field" datatype="key">editorctype</bcf:field>
- <bcf:field fieldtype="field" datatype="key">editortype</bcf:field>
- <bcf:field fieldtype="field" datatype="key">bookpagination</bcf:field>
- <bcf:field fieldtype="field" datatype="key">nameatype</bcf:field>
- <bcf:field fieldtype="field" datatype="key">namebtype</bcf:field>
- <bcf:field fieldtype="field" datatype="key">namectype</bcf:field>
- <bcf:field fieldtype="field" datatype="key">pagination</bcf:field>
- <bcf:field fieldtype="field" datatype="key">pubstate</bcf:field>
- <bcf:field fieldtype="field" datatype="key">type</bcf:field>
- <bcf:field fieldtype="list" datatype="key">language</bcf:field>
- <bcf:field fieldtype="list" datatype="key">origlanguage</bcf:field>
- <bcf:field fieldtype="field" datatype="entrykey">crossref</bcf:field>
- <bcf:field fieldtype="field" datatype="entrykey">xref</bcf:field>
- <bcf:field fieldtype="field" datatype="date" skip_output="true">date</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart" nullok="true">endyear</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart" nullok="true">year</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">month</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">day</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">hour</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">minute</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">second</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">timezone</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">season</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">endmonth</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">endday</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">endhour</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">endminute</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">endsecond</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">endtimezone</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">endseason</bcf:field>
- <bcf:field fieldtype="field" datatype="date" skip_output="true">eventdate</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart" nullok="true">eventendyear</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart" nullok="true">eventyear</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventmonth</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventday</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventhour</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventminute</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventsecond</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventtimezone</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventseason</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventendmonth</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventendday</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventendhour</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventendminute</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventendsecond</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventendtimezone</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">eventendseason</bcf:field>
- <bcf:field fieldtype="field" datatype="date" skip_output="true">origdate</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart" nullok="true">origendyear</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart" nullok="true">origyear</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origmonth</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origday</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">orighour</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origminute</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origsecond</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origtimezone</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origseason</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origendmonth</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origendday</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origendhour</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origendminute</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origendsecond</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origendtimezone</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">origendseason</bcf:field>
- <bcf:field fieldtype="field" datatype="date" skip_output="true">urldate</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart" nullok="true">urlendyear</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart" nullok="true">urlyear</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlmonth</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlday</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlhour</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlminute</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlsecond</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urltimezone</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlseason</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlendmonth</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlendday</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlendhour</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlendminute</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlendsecond</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlendtimezone</bcf:field>
- <bcf:field fieldtype="field" datatype="datepart">urlendseason</bcf:field>
- <bcf:field fieldtype="field" datatype="verbatim">doi</bcf:field>
- <bcf:field fieldtype="field" datatype="verbatim">eprint</bcf:field>
- <bcf:field fieldtype="field" datatype="verbatim">file</bcf:field>
- <bcf:field fieldtype="field" datatype="verbatim">verba</bcf:field>
- <bcf:field fieldtype="field" datatype="verbatim">verbb</bcf:field>
- <bcf:field fieldtype="field" datatype="verbatim">verbc</bcf:field>
- <bcf:field fieldtype="field" datatype="uri">url</bcf:field>
- <bcf:field fieldtype="field" format="xsv" datatype="entrykey" skip_output="true">xdata</bcf:field>
- <bcf:field fieldtype="field" format="xsv" datatype="entrykey" skip_output="true">ids</bcf:field>
- <bcf:field fieldtype="field" format="xsv" datatype="entrykey" skip_output="true">entryset</bcf:field>
- <bcf:field fieldtype="field" format="xsv" datatype="entrykey">related</bcf:field>
- <bcf:field fieldtype="field" format="xsv" datatype="keyword">keywords</bcf:field>
- <bcf:field fieldtype="field" format="xsv" datatype="option" skip_output="true">options</bcf:field>
- <bcf:field fieldtype="field" format="xsv" datatype="option" skip_output="true">relatedoptions</bcf:field>
- <bcf:field fieldtype="field" datatype="range">pages</bcf:field>
- <bcf:field fieldtype="field" datatype="code">execute</bcf:field>
- </bcf:fields>
- <bcf:entryfields>
- <bcf:field>abstract</bcf:field>
- <bcf:field>annotation</bcf:field>
- <bcf:field>authortype</bcf:field>
- <bcf:field>bookpagination</bcf:field>
- <bcf:field>crossref</bcf:field>
- <bcf:field>day</bcf:field>
- <bcf:field>doi</bcf:field>
- <bcf:field>eprint</bcf:field>
- <bcf:field>eprintclass</bcf:field>
- <bcf:field>eprinttype</bcf:field>
- <bcf:field>endday</bcf:field>
- <bcf:field>endhour</bcf:field>
- <bcf:field>endminute</bcf:field>
- <bcf:field>endmonth</bcf:field>
- <bcf:field>endseason</bcf:field>
- <bcf:field>endsecond</bcf:field>
- <bcf:field>endtimezone</bcf:field>
- <bcf:field>endyear</bcf:field>
- <bcf:field>entryset</bcf:field>
- <bcf:field>entrysubtype</bcf:field>
- <bcf:field>execute</bcf:field>
- <bcf:field>file</bcf:field>
- <bcf:field>gender</bcf:field>
- <bcf:field>hour</bcf:field>
- <bcf:field>ids</bcf:field>
- <bcf:field>indextitle</bcf:field>
- <bcf:field>indexsorttitle</bcf:field>
- <bcf:field>isan</bcf:field>
- <bcf:field>ismn</bcf:field>
- <bcf:field>iswc</bcf:field>
- <bcf:field>keywords</bcf:field>
- <bcf:field>label</bcf:field>
- <bcf:field>langid</bcf:field>
- <bcf:field>langidopts</bcf:field>
- <bcf:field>library</bcf:field>
- <bcf:field>lista</bcf:field>
- <bcf:field>listb</bcf:field>
- <bcf:field>listc</bcf:field>
- <bcf:field>listd</bcf:field>
- <bcf:field>liste</bcf:field>
- <bcf:field>listf</bcf:field>
- <bcf:field>minute</bcf:field>
- <bcf:field>month</bcf:field>
- <bcf:field>namea</bcf:field>
- <bcf:field>nameb</bcf:field>
- <bcf:field>namec</bcf:field>
- <bcf:field>nameatype</bcf:field>
- <bcf:field>namebtype</bcf:field>
- <bcf:field>namectype</bcf:field>
- <bcf:field>nameaddon</bcf:field>
- <bcf:field>options</bcf:field>
- <bcf:field>origday</bcf:field>
- <bcf:field>origendday</bcf:field>
- <bcf:field>origendhour</bcf:field>
- <bcf:field>origendminute</bcf:field>
- <bcf:field>origendmonth</bcf:field>
- <bcf:field>origendseason</bcf:field>
- <bcf:field>origendsecond</bcf:field>
- <bcf:field>origendtimezone</bcf:field>
- <bcf:field>origendyear</bcf:field>
- <bcf:field>orighour</bcf:field>
- <bcf:field>origminute</bcf:field>
- <bcf:field>origmonth</bcf:field>
- <bcf:field>origseason</bcf:field>
- <bcf:field>origsecond</bcf:field>
- <bcf:field>origtimezone</bcf:field>
- <bcf:field>origyear</bcf:field>
- <bcf:field>origlocation</bcf:field>
- <bcf:field>origpublisher</bcf:field>
- <bcf:field>origtitle</bcf:field>
- <bcf:field>pagination</bcf:field>
- <bcf:field>presort</bcf:field>
- <bcf:field>related</bcf:field>
- <bcf:field>relatedoptions</bcf:field>
- <bcf:field>relatedstring</bcf:field>
- <bcf:field>relatedtype</bcf:field>
- <bcf:field>season</bcf:field>
- <bcf:field>second</bcf:field>
- <bcf:field>shortauthor</bcf:field>
- <bcf:field>shorteditor</bcf:field>
- <bcf:field>shorthand</bcf:field>
- <bcf:field>shorthandintro</bcf:field>
- <bcf:field>shortjournal</bcf:field>
- <bcf:field>shortseries</bcf:field>
- <bcf:field>shorttitle</bcf:field>
- <bcf:field>sortkey</bcf:field>
- <bcf:field>sortname</bcf:field>
- <bcf:field>sortshorthand</bcf:field>
- <bcf:field>sorttitle</bcf:field>
- <bcf:field>sortyear</bcf:field>
- <bcf:field>timezone</bcf:field>
- <bcf:field>url</bcf:field>
- <bcf:field>urlday</bcf:field>
- <bcf:field>urlendday</bcf:field>
- <bcf:field>urlendhour</bcf:field>
- <bcf:field>urlendminute</bcf:field>
- <bcf:field>urlendmonth</bcf:field>
- <bcf:field>urlendsecond</bcf:field>
- <bcf:field>urlendtimezone</bcf:field>
- <bcf:field>urlendyear</bcf:field>
- <bcf:field>urlhour</bcf:field>
- <bcf:field>urlminute</bcf:field>
- <bcf:field>urlmonth</bcf:field>
- <bcf:field>urlsecond</bcf:field>
- <bcf:field>urltimezone</bcf:field>
- <bcf:field>urlyear</bcf:field>
- <bcf:field>usera</bcf:field>
- <bcf:field>userb</bcf:field>
- <bcf:field>userc</bcf:field>
- <bcf:field>userd</bcf:field>
- <bcf:field>usere</bcf:field>
- <bcf:field>userf</bcf:field>
- <bcf:field>verba</bcf:field>
- <bcf:field>verbb</bcf:field>
- <bcf:field>verbc</bcf:field>
- <bcf:field>xdata</bcf:field>
- <bcf:field>xref</bcf:field>
- <bcf:field>year</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>set</bcf:entrytype>
- <bcf:field>entryset</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>article</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>annotator</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>commentator</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editora</bcf:field>
- <bcf:field>editorb</bcf:field>
- <bcf:field>editorc</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>editoratype</bcf:field>
- <bcf:field>editorbtype</bcf:field>
- <bcf:field>editorctype</bcf:field>
- <bcf:field>eid</bcf:field>
- <bcf:field>issn</bcf:field>
- <bcf:field>issue</bcf:field>
- <bcf:field>issuetitle</bcf:field>
- <bcf:field>issuesubtitle</bcf:field>
- <bcf:field>issuetitleaddon</bcf:field>
- <bcf:field>journalsubtitle</bcf:field>
- <bcf:field>journaltitle</bcf:field>
- <bcf:field>journaltitleaddon</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>origlanguage</bcf:field>
- <bcf:field>pages</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>series</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>translator</bcf:field>
- <bcf:field>version</bcf:field>
- <bcf:field>volume</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>bibnote</bcf:entrytype>
- <bcf:field>note</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>book</bcf:entrytype>
- <bcf:field>author</bcf:field>
- <bcf:field>addendum</bcf:field>
- <bcf:field>afterword</bcf:field>
- <bcf:field>annotator</bcf:field>
- <bcf:field>chapter</bcf:field>
- <bcf:field>commentator</bcf:field>
- <bcf:field>edition</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editora</bcf:field>
- <bcf:field>editorb</bcf:field>
- <bcf:field>editorc</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>editoratype</bcf:field>
- <bcf:field>editorbtype</bcf:field>
- <bcf:field>editorctype</bcf:field>
- <bcf:field>eid</bcf:field>
- <bcf:field>foreword</bcf:field>
- <bcf:field>introduction</bcf:field>
- <bcf:field>isbn</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>maintitle</bcf:field>
- <bcf:field>maintitleaddon</bcf:field>
- <bcf:field>mainsubtitle</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>origlanguage</bcf:field>
- <bcf:field>pages</bcf:field>
- <bcf:field>pagetotal</bcf:field>
- <bcf:field>part</bcf:field>
- <bcf:field>publisher</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>series</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>translator</bcf:field>
- <bcf:field>volume</bcf:field>
- <bcf:field>volumes</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>mvbook</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>afterword</bcf:field>
- <bcf:field>annotator</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>commentator</bcf:field>
- <bcf:field>edition</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editora</bcf:field>
- <bcf:field>editorb</bcf:field>
- <bcf:field>editorc</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>editoratype</bcf:field>
- <bcf:field>editorbtype</bcf:field>
- <bcf:field>editorctype</bcf:field>
- <bcf:field>foreword</bcf:field>
- <bcf:field>introduction</bcf:field>
- <bcf:field>isbn</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>origlanguage</bcf:field>
- <bcf:field>pagetotal</bcf:field>
- <bcf:field>publisher</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>series</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>translator</bcf:field>
- <bcf:field>volume</bcf:field>
- <bcf:field>volumes</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>inbook</bcf:entrytype>
- <bcf:entrytype>bookinbook</bcf:entrytype>
- <bcf:entrytype>suppbook</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>afterword</bcf:field>
- <bcf:field>annotator</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>booktitle</bcf:field>
- <bcf:field>bookauthor</bcf:field>
- <bcf:field>booksubtitle</bcf:field>
- <bcf:field>booktitleaddon</bcf:field>
- <bcf:field>chapter</bcf:field>
- <bcf:field>commentator</bcf:field>
- <bcf:field>edition</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editora</bcf:field>
- <bcf:field>editorb</bcf:field>
- <bcf:field>editorc</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>editoratype</bcf:field>
- <bcf:field>editorbtype</bcf:field>
- <bcf:field>editorctype</bcf:field>
- <bcf:field>eid</bcf:field>
- <bcf:field>foreword</bcf:field>
- <bcf:field>introduction</bcf:field>
- <bcf:field>isbn</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>mainsubtitle</bcf:field>
- <bcf:field>maintitle</bcf:field>
- <bcf:field>maintitleaddon</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>origlanguage</bcf:field>
- <bcf:field>part</bcf:field>
- <bcf:field>publisher</bcf:field>
- <bcf:field>pages</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>series</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>translator</bcf:field>
- <bcf:field>volume</bcf:field>
- <bcf:field>volumes</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>booklet</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>chapter</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>eid</bcf:field>
- <bcf:field>howpublished</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>pages</bcf:field>
- <bcf:field>pagetotal</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>type</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>collection</bcf:entrytype>
- <bcf:entrytype>reference</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>afterword</bcf:field>
- <bcf:field>annotator</bcf:field>
- <bcf:field>chapter</bcf:field>
- <bcf:field>commentator</bcf:field>
- <bcf:field>edition</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editora</bcf:field>
- <bcf:field>editorb</bcf:field>
- <bcf:field>editorc</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>editoratype</bcf:field>
- <bcf:field>editorbtype</bcf:field>
- <bcf:field>editorctype</bcf:field>
- <bcf:field>eid</bcf:field>
- <bcf:field>foreword</bcf:field>
- <bcf:field>introduction</bcf:field>
- <bcf:field>isbn</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>mainsubtitle</bcf:field>
- <bcf:field>maintitle</bcf:field>
- <bcf:field>maintitleaddon</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>origlanguage</bcf:field>
- <bcf:field>pages</bcf:field>
- <bcf:field>pagetotal</bcf:field>
- <bcf:field>part</bcf:field>
- <bcf:field>publisher</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>series</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>translator</bcf:field>
- <bcf:field>volume</bcf:field>
- <bcf:field>volumes</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>mvcollection</bcf:entrytype>
- <bcf:entrytype>mvreference</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>afterword</bcf:field>
- <bcf:field>annotator</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>commentator</bcf:field>
- <bcf:field>edition</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editora</bcf:field>
- <bcf:field>editorb</bcf:field>
- <bcf:field>editorc</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>editoratype</bcf:field>
- <bcf:field>editorbtype</bcf:field>
- <bcf:field>editorctype</bcf:field>
- <bcf:field>foreword</bcf:field>
- <bcf:field>introduction</bcf:field>
- <bcf:field>isbn</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>origlanguage</bcf:field>
- <bcf:field>publisher</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>translator</bcf:field>
- <bcf:field>volume</bcf:field>
- <bcf:field>volumes</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>incollection</bcf:entrytype>
- <bcf:entrytype>suppcollection</bcf:entrytype>
- <bcf:entrytype>inreference</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>afterword</bcf:field>
- <bcf:field>annotator</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>booksubtitle</bcf:field>
- <bcf:field>booktitle</bcf:field>
- <bcf:field>booktitleaddon</bcf:field>
- <bcf:field>chapter</bcf:field>
- <bcf:field>commentator</bcf:field>
- <bcf:field>edition</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editora</bcf:field>
- <bcf:field>editorb</bcf:field>
- <bcf:field>editorc</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>editoratype</bcf:field>
- <bcf:field>editorbtype</bcf:field>
- <bcf:field>editorctype</bcf:field>
- <bcf:field>eid</bcf:field>
- <bcf:field>foreword</bcf:field>
- <bcf:field>introduction</bcf:field>
- <bcf:field>isbn</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>mainsubtitle</bcf:field>
- <bcf:field>maintitle</bcf:field>
- <bcf:field>maintitleaddon</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>origlanguage</bcf:field>
- <bcf:field>pages</bcf:field>
- <bcf:field>part</bcf:field>
- <bcf:field>publisher</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>series</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>translator</bcf:field>
- <bcf:field>volume</bcf:field>
- <bcf:field>volumes</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>dataset</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>edition</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>organization</bcf:field>
- <bcf:field>publisher</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>series</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>type</bcf:field>
- <bcf:field>version</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>manual</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>chapter</bcf:field>
- <bcf:field>edition</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>eid</bcf:field>
- <bcf:field>isbn</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>organization</bcf:field>
- <bcf:field>pages</bcf:field>
- <bcf:field>pagetotal</bcf:field>
- <bcf:field>publisher</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>series</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>type</bcf:field>
- <bcf:field>version</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>misc</bcf:entrytype>
- <bcf:entrytype>software</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>howpublished</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>organization</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>type</bcf:field>
- <bcf:field>version</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>online</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>organization</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>version</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>patent</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>holder</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>type</bcf:field>
- <bcf:field>version</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>periodical</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editora</bcf:field>
- <bcf:field>editorb</bcf:field>
- <bcf:field>editorc</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>editoratype</bcf:field>
- <bcf:field>editorbtype</bcf:field>
- <bcf:field>editorctype</bcf:field>
- <bcf:field>issn</bcf:field>
- <bcf:field>issue</bcf:field>
- <bcf:field>issuesubtitle</bcf:field>
- <bcf:field>issuetitle</bcf:field>
- <bcf:field>issuetitleaddon</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>season</bcf:field>
- <bcf:field>series</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>volume</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>mvproceedings</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>eventday</bcf:field>
- <bcf:field>eventendday</bcf:field>
- <bcf:field>eventendhour</bcf:field>
- <bcf:field>eventendminute</bcf:field>
- <bcf:field>eventendmonth</bcf:field>
- <bcf:field>eventendseason</bcf:field>
- <bcf:field>eventendsecond</bcf:field>
- <bcf:field>eventendtimezone</bcf:field>
- <bcf:field>eventendyear</bcf:field>
- <bcf:field>eventhour</bcf:field>
- <bcf:field>eventminute</bcf:field>
- <bcf:field>eventmonth</bcf:field>
- <bcf:field>eventseason</bcf:field>
- <bcf:field>eventsecond</bcf:field>
- <bcf:field>eventtimezone</bcf:field>
- <bcf:field>eventyear</bcf:field>
- <bcf:field>eventtitle</bcf:field>
- <bcf:field>eventtitleaddon</bcf:field>
- <bcf:field>isbn</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>organization</bcf:field>
- <bcf:field>pagetotal</bcf:field>
- <bcf:field>publisher</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>series</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>venue</bcf:field>
- <bcf:field>volumes</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>proceedings</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>chapter</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>eid</bcf:field>
- <bcf:field>eventday</bcf:field>
- <bcf:field>eventendday</bcf:field>
- <bcf:field>eventendhour</bcf:field>
- <bcf:field>eventendminute</bcf:field>
- <bcf:field>eventendmonth</bcf:field>
- <bcf:field>eventendseason</bcf:field>
- <bcf:field>eventendsecond</bcf:field>
- <bcf:field>eventendtimezone</bcf:field>
- <bcf:field>eventendyear</bcf:field>
- <bcf:field>eventhour</bcf:field>
- <bcf:field>eventminute</bcf:field>
- <bcf:field>eventmonth</bcf:field>
- <bcf:field>eventseason</bcf:field>
- <bcf:field>eventsecond</bcf:field>
- <bcf:field>eventtimezone</bcf:field>
- <bcf:field>eventyear</bcf:field>
- <bcf:field>eventtitle</bcf:field>
- <bcf:field>eventtitleaddon</bcf:field>
- <bcf:field>isbn</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>mainsubtitle</bcf:field>
- <bcf:field>maintitle</bcf:field>
- <bcf:field>maintitleaddon</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>organization</bcf:field>
- <bcf:field>pages</bcf:field>
- <bcf:field>pagetotal</bcf:field>
- <bcf:field>part</bcf:field>
- <bcf:field>publisher</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>series</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>venue</bcf:field>
- <bcf:field>volume</bcf:field>
- <bcf:field>volumes</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>inproceedings</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>booksubtitle</bcf:field>
- <bcf:field>booktitle</bcf:field>
- <bcf:field>booktitleaddon</bcf:field>
- <bcf:field>chapter</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editortype</bcf:field>
- <bcf:field>eid</bcf:field>
- <bcf:field>eventday</bcf:field>
- <bcf:field>eventendday</bcf:field>
- <bcf:field>eventendhour</bcf:field>
- <bcf:field>eventendminute</bcf:field>
- <bcf:field>eventendmonth</bcf:field>
- <bcf:field>eventendseason</bcf:field>
- <bcf:field>eventendsecond</bcf:field>
- <bcf:field>eventendtimezone</bcf:field>
- <bcf:field>eventendyear</bcf:field>
- <bcf:field>eventhour</bcf:field>
- <bcf:field>eventminute</bcf:field>
- <bcf:field>eventmonth</bcf:field>
- <bcf:field>eventseason</bcf:field>
- <bcf:field>eventsecond</bcf:field>
- <bcf:field>eventtimezone</bcf:field>
- <bcf:field>eventyear</bcf:field>
- <bcf:field>eventtitle</bcf:field>
- <bcf:field>eventtitleaddon</bcf:field>
- <bcf:field>isbn</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>mainsubtitle</bcf:field>
- <bcf:field>maintitle</bcf:field>
- <bcf:field>maintitleaddon</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>organization</bcf:field>
- <bcf:field>pages</bcf:field>
- <bcf:field>part</bcf:field>
- <bcf:field>publisher</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>series</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>venue</bcf:field>
- <bcf:field>volume</bcf:field>
- <bcf:field>volumes</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>report</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>chapter</bcf:field>
- <bcf:field>eid</bcf:field>
- <bcf:field>institution</bcf:field>
- <bcf:field>isrn</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>number</bcf:field>
- <bcf:field>pages</bcf:field>
- <bcf:field>pagetotal</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>type</bcf:field>
- <bcf:field>version</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>thesis</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>chapter</bcf:field>
- <bcf:field>eid</bcf:field>
- <bcf:field>institution</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>pages</bcf:field>
- <bcf:field>pagetotal</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>type</bcf:field>
- </bcf:entryfields>
- <bcf:entryfields>
- <bcf:entrytype>unpublished</bcf:entrytype>
- <bcf:field>addendum</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>eventday</bcf:field>
- <bcf:field>eventendday</bcf:field>
- <bcf:field>eventendhour</bcf:field>
- <bcf:field>eventendminute</bcf:field>
- <bcf:field>eventendmonth</bcf:field>
- <bcf:field>eventendseason</bcf:field>
- <bcf:field>eventendsecond</bcf:field>
- <bcf:field>eventendtimezone</bcf:field>
- <bcf:field>eventendyear</bcf:field>
- <bcf:field>eventhour</bcf:field>
- <bcf:field>eventminute</bcf:field>
- <bcf:field>eventmonth</bcf:field>
- <bcf:field>eventseason</bcf:field>
- <bcf:field>eventsecond</bcf:field>
- <bcf:field>eventtimezone</bcf:field>
- <bcf:field>eventyear</bcf:field>
- <bcf:field>eventtitle</bcf:field>
- <bcf:field>eventtitleaddon</bcf:field>
- <bcf:field>howpublished</bcf:field>
- <bcf:field>language</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>pubstate</bcf:field>
- <bcf:field>subtitle</bcf:field>
- <bcf:field>title</bcf:field>
- <bcf:field>titleaddon</bcf:field>
- <bcf:field>type</bcf:field>
- <bcf:field>venue</bcf:field>
- </bcf:entryfields>
- <bcf:multiscriptfields>
- <bcf:field>abstract</bcf:field>
- <bcf:field>addendum</bcf:field>
- <bcf:field>afterword</bcf:field>
- <bcf:field>annotator</bcf:field>
- <bcf:field>author</bcf:field>
- <bcf:field>bookauthor</bcf:field>
- <bcf:field>booksubtitle</bcf:field>
- <bcf:field>booktitle</bcf:field>
- <bcf:field>booktitleaddon</bcf:field>
- <bcf:field>chapter</bcf:field>
- <bcf:field>commentator</bcf:field>
- <bcf:field>editor</bcf:field>
- <bcf:field>editora</bcf:field>
- <bcf:field>editorb</bcf:field>
- <bcf:field>editorc</bcf:field>
- <bcf:field>foreword</bcf:field>
- <bcf:field>holder</bcf:field>
- <bcf:field>institution</bcf:field>
- <bcf:field>introduction</bcf:field>
- <bcf:field>issuesubtitle</bcf:field>
- <bcf:field>issuetitle</bcf:field>
- <bcf:field>issuetitleaddon</bcf:field>
- <bcf:field>journalsubtitle</bcf:field>
- <bcf:field>journaltitle</bcf:field>
- <bcf:field>journaltitleaddon</bcf:field>
- <bcf:field>location</bcf:field>
- <bcf:field>mainsubtitle</bcf:field>
- <bcf:field>maintitle</bcf:field>
- <bcf:field>maintitleaddon</bcf:field>
- <bcf:field>nameaddon</bcf:field>
- <bcf:field>note</bcf:field>
- <bcf:field>organization</bcf:field>
- <bcf:field>origlanguage</bcf:field>
- <bcf:field>origlocation</bcf:field>
- <bcf:field>origpublisher</bcf:field>
- <bcf:field>origtitle</bcf:field>
- <bcf:field>part</bcf:field>
- <bcf:field>publisher</bcf:field>
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- <bcf:citekey order="4">ncgwalter</bcf:citekey>
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- <bcf:citekey order="8">heatkernel</bcf:citekey>
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diff --git a/src/thesis/main.log b/src/thesis/main.log
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