Added the continuous functional calculus.

src/op/c-star/cont.tex

@@ -0,0 +1,65 @@
+\section{The Continuous Functional Calculus}
+\label{section:continuous-functional-calculus}
+
+\begin{theorem}[Spectral Theorem for $C^*$-Algebras]
+\label{theorem:spectral-c-star}
+    Let $A$ be a unital $C^*$-algebra and $x \in A$ be normal, then the mapping
+    \[
+    \Phi: \Omega(A[x]) \to \sigma_A(x) \quad \Phi(\psi) = \Gamma_{A[x]}(x)(\psi) = \psi(x)
+    \]
+
+    is a homeomorphism. 
+\end{theorem}
+\begin{proof}
+    Firstly, $A[x]$ is commutative by \autoref{proposition:generated-subalgebra-dense}. Thus \autoref{corollary:c-star-algebra-preserve-spectrum} and (3) of \autoref{proposition:gelfand-transform-gymnastics} imply that
+    \[
+    \Phi(\Omega(A[x])) = \Gamma_{A[x]}(\Omega(A[x])) = \sigma_{A[x]}(x) = \sigma_A(x)
+    \]
+
+    and $\Phi$ is a surjection onto $\sigma_A(x)$. 
+
+    On the other hand, the \hyperref[Gelfand-Naimark Theorem]{theorem:gelfand-naimark} implies that $\Phi(x^*) = \ol{\Phi(x)}$, so since $A[x]$ is the smallest $C^*$-algebra containing $x$, any element $\psi \in \Omega(A[x])$ is uniquely determined by $\psi(x)$. Therefore $\Phi$ is injective. 
+
+    Finally, since $\Omega(A[x])$ is equipped with the weak* topology and $\Phi$ is the evaluation map at $x$, it is continuous. 
+
+    By \autoref{proposition:compact-hausdorff-homeomorphism}, $\Phi$ is a homeomorphism. 
+\end{proof}
+
+\begin{definition}[Continuous Functional Calculus]
+\label{definition:continuous-functional-calculus}
+    Let $A$ be a unital $C^*$-algebra and $x \in A$ be normal, then there exists a unique continuous unital *-homomorphism
+    \[
+    C(\sigma_A(x); \complex) \to A[x] \quad f \mapsto f(x)
+    \]
+
+    such that:
+    \begin{enumerate}
+        \item $\one(x) = 1_A$. 
+        \item $\text{Id}(x) = x$.
+        \item $\overline{\text{Id}}(x) = x^*$.
+    \end{enumerate}
+
+    Moreover,
+    \begin{enumerate}[start=3]
+        \item Under the identification $\Omega(A[x]) = \sigma_A(x)$, for each $f \in C(\sigma_A(x); \complex)$, $f(x) = \Gamma^{-1}_{A[x]}(f)$.
+        \item The mapping $f \mapsto f(x)$ is a unital *-isomorphism.  
+    \end{enumerate}
+    
+\end{definition}
+\begin{proof}
+    (1), (4), (5): By the \hyperref[Spectral Theorem]{theorem:spectral-c-star}, $\Omega(A[x])$ and $\sigma_A(x)$ may be identified. For each $f \in C(\sigma_A(x); \complex)$, define $f(x) = \Gamma^{-1}_{A[x]}(f)$, then the mapping $f \mapsto f(x)$ is a unital *-isomorphism by the \hyperref[Gelfand-Naimark Theorem]{theorem:gelfand-naimark}. 
+
+    (2): The identification 
+    \[
+    \Phi: \Omega(A[x]) \to \sigma_A(x) \quad \Phi(\psi) = \Gamma_{A[x]}(x)(\psi) = \psi(x)
+    \]
+
+    given by the Spectral Theorem implies that $\Gamma_{A[x]}(x) = \text{Id}$.
+
+    (3): The mapping $f \mapsto f(x)$ is a *-homomorphism. 
+
+    (Uniqueness): By the \hyperref[Stone-Weierstrass Theorem]{theorem:complex-stone-weierstrass}.
+\end{proof}
+
+
+

src/op/c-star/index.tex

@@ -2,8 +2,10 @@
 \label{chap:c-star-algebras}
 
 \input{./involution.tex}
+\input{./sub.tex}
 \input{./unitary.tex}
 \input{./sa.tex}
 \input{./order.tex}
 \input{./homomorphism.tex}
-\input{./gelfand.tex}
+\input{./gelfand.tex}
+\input{./cont.tex}

src/op/c-star/involution.tex

@@ -41,17 +41,4 @@
     (3): For any $x \in A$, $(x^{-1})^*x^* = (x^{-1}x)^* = 1 = (xx^{-1})^* = x^*(x^{-1})^*$.
 \end{proof}
 
-\begin{definition}[*-Homomorphism]
-\label{definition:star-homomorphism}
-    Let $A, B$ be $C^*$-algebras and $\phi: A \to B$, then $\phi$ is a \textbf{*-homomorphism} if:
-    \begin{enumerate}
-        \item $\phi$ is a homomorphism of Banach algebras.
-        \item For every $x \in A$, $\phi(x^*) = \phi(x)^*$. 
-    \end{enumerate}
-
-    If in addition, $\phi(1) = 1$, then $\phi$ is a \textbf{unital *-homomorphism}. 
-\end{definition}
-
-
-
 

src/op/c-star/sub.tex

@@ -0,0 +1,41 @@
+\section{Subalgebras}
+\label{section:sub-c-star-algebras}
+
+\begin{definition}[Generated Subalgebra]
+\label{definition:generated-subalgebra}
+    Let $A$ be a unital $C^*$-algebra and $S \subset A$, then $A[S]$ is the smallest $C^*$-subalgebra of $A$ containing $1$ and $S$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:generated-subalgebra-dense}
+    Let $A$ be a unital $C^*$-algebra, $S \subset A$, and $\mathcal{S} = S \cup \bracs{x^*|x \in S}$, then
+    \begin{enumerate}
+        \item The linear span
+        \[
+        \text{span}\bracs{\prod_{j = 1}^n x_j \bigg | \seqf{x_j} \subset \mathcal{S}}
+        \]
+
+        is dense in $A[S]$. 
+        \item If for any $x, y \in \mathcal{S}$, $xy = yx$, then $A[S]$ is commutative.
+        \item For any normal element $x \in A$, $A[x]$ is commutative. 
+    \end{enumerate}
+\end{proposition}
+% Obvious so omitted. 
+
+\begin{proposition}
+\label{proposition:c-star-algebra-preserve-gl}
+    Let $A$ be a unital $C^*$-algebra, $B \subset A$ be a $C^*$-subalgebra containing $1$, then $G(B) = G(A) \cap B$. 
+\end{proposition}
+\begin{proof}
+    Let $x \in G(A) \cap B$, then $x^*x \in G(A) \cap B$ as well. In particular, $0 \not\in \sigma_A(x^*x)$. Since $x^*x \in A_{sa}$, $\sigma_A(x^*x) \subset \real$ by \autoref{proposition:self-adjoint-spectrum}. By \autoref{proposition:spectrum-subalgebra-gymnastics}, $\partial \sigma_B(x^*x) \subset \sigma_A(x^*x) \subset \real$. Thus $\sigma_B(x^*x) \subset \real$ as well, which means that $\partial \sigma_B(x^*x) = \sigma_B(x^*x) = \sigma_A(x^*x)$. Therefore $0 \not\in \sigma_A(x^*x) = \sigma_B(x^*x)$, $x^*x \in G(B)$, and $x \in G(B)$. 
+\end{proof}
+
+\begin{corollary}
+\label{corollary:c-star-algebra-preserve-spectrum}
+    Let $A$ be a unital $C^*$-algebra, $B \subset A$ be a $C^*$-subalgebra containing $1$, and $x \in B$, then $\sigma_A(x) = \sigma_B(x)$. 
+\end{corollary}
+\begin{proof}
+    By \autoref{proposition:c-star-algebra-preserve-gl}. 
+\end{proof}
+
+