Added first draft of the Borel functional calculus.

src/op/vn/fc.tex

@@ -60,7 +60,7 @@
         \item Composition and transposition are continuous in the operator norm on $B(H)$
     \end{enumerate}
 
-    the map $I_E$ restricted to $B^\infty(X; \complex)$ is a *-homomorphism by continuity. By \hyperref[Goldstine's Theorem]{theorem:goldstine-weak}, $C(X; \complex) \subset B^\infty(X; \complex)$ is weak*-dense in $C(X; \complex)^{**}$. So as
+    the map $I_E$ restricted to $B^\infty(X; \complex)$ is a *-homomorphism by continuity. By \hyperref[Goldstine's Theorem]{corollary:weak-dense-unit-ball}, $C(X; \complex) \subset B^\infty(X; \complex)$ is weak*-dense in $C(X; \complex)^{**}$. So as
     \begin{enumerate}[label=(\roman*)]
         \item $I_E$ restricted to $B^\infty(X; \complex)$ is a *-homomorphism. 
         \item The involution $\phi \mapsto \ol \phi$ is weak*-continuous on $C(X; \complex)^{**}$. 
@@ -149,3 +149,33 @@
     B = \bracs{\int_{\Omega(A)} \phi dE \bigg | \phi \in C(\Omega(A); \complex)^{**}}
     \]
 \end{proof}
+
+\begin{definition}[Borel Functional Calculus]
+\label{definition:borel-functional-calculus}
+    Let $H$ be a complex Hilbert space, $T \in B(H)$ be normal, and $A \subset B(H)$ be the smallest von Neumann algebra acting on $H$ containing $T$ and $I$, then there exists a unique continuous unital *-homomorphism
+    \[
+    C(\sigma_{B(H)}(T); \complex)^{**} \to A \quad \phi \mapsto \phi(T)
+    \]
+
+    such that:
+    \begin{enumerate}
+        \item $\one(T) = I$, $\text{Id}(T) = T$, $\ol{Id}(T) = T^*$. 
+        \item The mapping $\phi \mapsto \phi(T)$ is continuous from the weak* topology on $C(\sigma_{B(H)}(T); \complex)^{**}$ to the weak operator topology on $C(\sigma_{B(H)}(T); \complex)$. 
+    \end{enumerate}
+
+    Moreover, there exists a unique spectral measure $E: \sigma_{B(H)}(T) \to A$
+\end{definition}
+\begin{proof}
+    By the \hyperref[Spectral Theorem]{theorem:spectral-theorem-vn-1} applied to $B(H)[T]$, there exists a unique spectral measure $E$ on $\sigma_{B(H)}(T)$ such that the mapping
+    \[
+    I_E: C(\sigma_{B(H)}(T); \complex)^{**} \to A \quad \phi \mapsto \int_{\sigma_A(T)} \phi dE
+    \]
+
+    extends the inverse Gelfand transform $\Gamma_{B(H)[T]}^{-1}: C(\sigma_{B(H)}(T); \complex) \to B(H)[T]$. 
+    
+    For each $\phi \in C(\sigma_{B(H)}; \complex)^{**}$, let $\phi(T) = \int_{\sigma_{B(H)}(T)}\phi dE$, then the mapping $\phi \mapsto \phi(T)$ is continuous from the weak* topology on $C(\sigma_{B(H)}(T); \complex)^{**}$ to the weak operator topology on $C(\sigma_{B(H)}(T); \complex)$ by \autoref{definition:spectral-measure-integral}.
+
+    Finally, by uniqueness of the \hyperref[continuous functional calculus]{definition:continuous-functional-calculus}, \hyperref[Goldstine's Theorem]{corollary:weak-dense-unit-ball}, and (2), the mapping $\phi \mapsto \phi(T)$ is unique. 
+\end{proof}
+
+