commit fed8832ba997fdca91d156fd6d92bcedf2fd0066
parent 8c28aa4d3a8944a30bedf77ac70e16615c906f7b
Author: miksa234 <milutin@popovic.xyz>
Date: Fri, 19 Feb 2021 19:35:17 +0100
mathematical incorenctness of discreet and discrete
Diffstat:
3 files changed, 8 insertions(+), 8 deletions(-)
diff --git a/week1.pdf b/week1.pdf
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diff --git a/week1.tex b/week1.tex
@@ -52,7 +52,7 @@
$A \times A \rightarrow A$ \hspace{0.1\textwidth} \textit{bilinear} \\
$(a, b)\ \mapsto \ a\cdot b$
\item
- $1a = a1 =a$ \hspace{0.08\textwidth} \textit{unitary} \\
+ $1a = a1 =a$ \hspace{0.08\textwidth} \textit{unital} \\
\end{enumerate}
\end{definition}
@@ -66,7 +66,7 @@
$(a^*)^* = a$ \hspace{0.1\textwidth} \textit{closure}
\end{enumerate}
\end{definition}
-In the following all unitary algebras are referred to as algebras.
+In the following all unital algebras are referred to as algebras.
\subsubsection{Functions on Discrete Spaces}
Let $X$ be a \textit{discretized topological} space with $N$ points.
@@ -129,7 +129,7 @@ Under the pointwise product:
\begin{question}
$\phi$ is in most cases not bijective, so how can we prove that there exists such a
- pullback for every map between discreet spaces which preserves information? For bijective
+ pullback for every map between discrete spaces which preserves information? For bijective
it is given by its inverse, which by definition exists because $\phi$ is a map.
Or I didn't understand this correctly?
\end{question}
@@ -163,7 +163,7 @@ Under the pointwise product:
a matrix algebra
\end{definition}
-So from a topological discreet space $X$, we can construct a $*$-algebra $C(X)$ which is isomorphic
+So from a topological discrete space $X$, we can construct a $*$-algebra $C(X)$ which is isomorphic
to a matrix algebra $A$. The question is can we construct $X$ given $A$? $A$ is a matrix algebra,
which are in most cases is not commutative, so the answer is generally no.
@@ -178,7 +178,7 @@ Or we can allow more morphisms(isomorphisms) between matrix algebras.
Exactly.
\subsubsection{Finite Inner Product Spaces and Representations}
-Until now we looked at a finite topological discreet space, moreover we can consider a
+Until now we looked at a finite topological discrete space, moreover we can consider a
finite dimensional inner product space $H$ (finite Hilbert-spaces), with inner product
$(\cdot,\cdot)\rightarrow \mathbb{C}$. $L(H)$ is the $*$-algebra of operators on $H$
with product given by composition and involution given by the adjoint, $T \mapsto T^*$.
@@ -215,7 +215,7 @@ Examples for reducible and irreducible representations
\begin{definition}
Let $(H_1, \pi _1)$ and $(H_2, \pi _2)$ be representations of a $*$-algebra $A$. They are called
- \textit{unitarily equivalent} if there exists a map $U: H_1 \rightarrow H_2$ such that.
+ \textit{unitary equivalent} if there exists a map $U: H_1 \rightarrow H_2$ such that.
\begin{align*}
\pi _1(a) = U^* \pi _2(a) U
\end{align*}
@@ -252,7 +252,7 @@ More on that in later chapters.
\begin{solution}
1. To show that $\pi (A)'$ is a $*$-algebra we have to show that it is unital, associative and involute.
And note that $\pi (a) \in L(H)\ \forall a \in A$.
- Unity is given by the unitary operator of the $*$-algebra of operators $L(H)$, which exists by definition
+ Unitarity is given by the unital operator of the $*$-algebra of operators $L(H)$, which exists by definition
because H is a inner product space. Associativity is given by $*$-algebra of $L(H)$, $L(H) \times L(H) \mapsto L(H)$,
which is associative by definition. Involutnes is also given by the $*$-algebra $L(H)$
with a map $*: L(H) \mapsto L(H)$ only for $T$ that commute with $\pi (a)$.
@@ -292,7 +292,7 @@ More on that in later chapters.
irreducible representation are of the form
$\pi _i:(\lambda_1,...,\lambda_N)\in \mathbb{C}^N \mapsto \lambda_i \in \mathbb{C}$ \\
for $i = 1,...,N \Rightarrow \hat{A} \simeq \{1,...,N\}.$
- \item Conclusion is that there is a duality between discreet spaces and commutative matrix algebra
+ \item Conclusion is that there is a duality between discrete spaces and commutative matrix algebra
this duality is called the \textit{finite dimensional Gelfand duality}
\end{itemize}
diff --git a/week2.pdf b/week2.pdf
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