First draft of spectral theorem II.

src/op/vn/fc.tex

@@ -23,6 +23,7 @@
     Let $X$ be a compact Hausdorff space, $H$ be a complex Hilbert space, and $E: \cb_X \to B(H)$ be a spectral measure relative to $H$, then:
     \begin{enumerate}
         \item For each $x, y \in H$, $\norm{E_{x, y}}_{\text{var}} \le \norm{x}_H \norm{y}_H$. 
+        \item For each $x \in H$, $E_{x, x}$ is positive.
     \end{enumerate}
 
     Let $\mathscr{E} \subset M_R(X; \complex)$ be the closed subspace generated by $\bracsn{E_{x, y}|x, y \in H}$, then
@@ -47,7 +48,9 @@
 
     As the above holds for all finite sequences of disjoint Borel sets, $\norm{E_{x, y}}_{\text{var}} \le \norm{x}_H \norm{y}_H$.
 
-    (2): For each $x, y \in H$ and $B, C \in \cb_X$, 
+    (2): For each $B \in \cb_X$, $E(B)$ is a projection, so $E_{x, x}(B) = \dpn{E(B)x, x}{H} \ge 0$.
+
+    (3): For each $x, y \in H$ and $B, C \in \cb_X$, 
     \[
     \int_C \one_B dE_{x, y} = \dpn{E(C \cap B)x, y}{H} = \dpn{E(C)E(B)x, y}{H} = E_{E(B)x, y}(C)
     \]
@@ -56,7 +59,7 @@
     
     Finally, let $\mu \in \mathscr{E}$ and $\nu \in M_R(X; \complex)$ with $\nu \ll \mu$, then by the \hyperref[Radon-Nikodym Theorem]{theorem:lebesgue-radon-nikodym}, there exists $f \in L^1(\mu; \complex)$ such that $d\nu = f d\mu \in \mathscr{E}$. 
 
-    (3): By (2), for any $\mu \in \mathscr{E}$ and $f \in L^1(\mu; \complex)$, $fd\mu \in \mathscr{E}$ as well. By \autoref{proposition:measures-dual-algebra}, there exists a unique weak*-continuous involution and separately weak*-continuous product on $\mathscr{E}^*$ making $\mathscr{E}^*$ a commutative unital $C^*$-algebra, and $J|_{C(X; \complex)}$ a unital *-homomorphism. Since
+    (4): By (3), for any $\mu \in \mathscr{E}$ and $f \in L^1(\mu; \complex)$, $fd\mu \in \mathscr{E}$ as well. By \autoref{proposition:measures-dual-algebra}, there exists a unique weak*-continuous involution and separately weak*-continuous product on $\mathscr{E}^*$ making $\mathscr{E}^*$ a commutative unital $C^*$-algebra, and $J|_{C(X; \complex)}$ a unital *-homomorphism. Since
     \begin{enumerate}[label=(\roman*)]
         \item $J$ is $\sigma(B^\infty(X; \complex), M_R(X; \complex))$-$\sigma(\mathscr{E}^*, \mathscr{E})$ continuous. 
         \item Conjugation on $B^\infty(X; \complex)$ is $\sigma(B^\infty(X; \complex), M_R(X; \complex))$-continuous.
@@ -141,7 +144,7 @@
         is a *-isomorphism. 
     \end{enumerate}
     
-    The mapping $\mathscr{E}^* \to B$ defined by $\phi \mapsto \int \phi dE$ is the \textbf{extended inverse Gelfand transform} of $A$. 
+    The measure $E$ is the \textbf{spectral measure associated with $A$}, and the homomorphism $I_E$ is the \textbf{extended inverse Gelfand transform} of $A$. 
 \end{theorem}
 \begin{proof}[Proof, {{\cite[Theorem 20.2]{Zhu}}}. ]
     (1): By the \hyperref[Gelfand-Naimark Theorem]{theorem:gelfand-naimark}, $\Gamma_A: A \to C(\Omega(A); \complex)$ is a *-isomorphism. For each $x, y \in H$, $\Gamma_A^{-1}$ induces a mapping
@@ -194,7 +197,85 @@
     On the other hand, by the \hyperref[Banach-Alaoglu Theorem]{theorem:alaoglu}, $\ol{B_{\mathscr{E}^*}(0, 1)}$ is weak*-compact, so $I_E(\ol{B_{\mathscr{E}^*}(0, 1)})$ is weak-operator compact. As $\Gamma_A: A \to C(\Omega(A); \complex)$ is an isometric isomorphism, $I_E(\ol{B_{\mathscr{E}^*}(0, 1)}) \supset \ol{B_A(0, 1)}$. By the \hyperref[Kaplansky Density Theorem]{theorem:kaplansky-density}, $I_E(\ol{B_{\mathscr{E}^*}(0, 1)}) \supset B_B(0, 1)$, and $I_E(\mathscr{E}^*) = B$.
 \end{proof}
 
-\begin{definition}[Borel Functional Calculus]
+\begin{theorem}[Spectral Theorem II]
+\label{theorem:spectral-theorem-vn-2}
+    Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a commutative $C^*$-subalgebra with $I \in A$, $B$ be the von Neumann algebra generated by $A$, $E: \cb_{\Omega(A)} \to B(H)$ be the spectral measure associated with $A$, $\mathscr{E} \subset M_R(\Omega(A); \complex)$ be the closed subspace generated by $\bracsn{E_{x, y}|x, y \in H}$, and $\seqi{\xi} \subset H$ be a maximal family such that the subspaces $\bracsn{A\xi_i|i \in I}$ are mutually orthogonal, then:
+    \begin{enumerate}
+        \item For each $i \in I$, there exists a finite positive Radon measure $\mu_i$ on $\Omega(A)$ such that for every Borel set $C \in \cb_{\Omega(A)}$, $\mu_i(C) = 0$ if and only if $E_{x, y}(C) = 0$ for all $x, y \in \ol{A\xi_i}$. Moreover, the measures $\seqi{\mu}$ are mutually singular. 
+        \item $\mathscr{E} = [l^1(I); L^1(\mu_i; \complex)]$ and $\mathscr{E}^* = [l^\infty(I); L^\infty(\mu_i; \complex)]$. 
+        \item There exists a unitary equivalence $U: H \to [l^2(I); L^2(\mu_i; \complex)]$ between $[l^\infty(I); L^\infty(\mu_i; \complex)]$ acting on $[l^2(I); L^2(\mu_i; \complex)]$ and $B$ acting on $H$, such that for each $i \in I$, $U|_{A\xi_i}$ is an isometry onto the $i$-th factor of $[l^2(I); L^2(\mu_i; \complex)]$. 
+    \end{enumerate}
+\end{theorem}
+\begin{proof}[Proof, {{\cite[Theorem 1.47]{FollandHarmonic}}}. ]
+    (1): Fix $ i \in I$ and let $\mu_i = E_{\xi_i, \xi_i}$, then for any $C \in \cb_{\Omega(A)}$ with $E_{x, y}(C) = 0$ for all $x, y \in \ol{A\xi_i}$, $\mu_i(C) = 0$. By (1) and (2) of \autoref{lemma:spectral-measure-properties}, $\mu_i$ is a finite positive Radon measure. 
+    
+    By \hyperref[Spectral Theorem I]{theorem:spectral-theorem-vn-1}, for each $S, T \in A$ and $C \in \cb_{\Omega(A)}$, 
+    \[
+    \dpn{E(C)S\xi_i, T\xi_i}{H} = \int_C \Gamma_AS \cdot \ol{\Gamma_AT} dE_{\xi_i, \xi_i}
+    \]
+
+    so $\Gamma_AS \cdot \ol{\Gamma_AT}dE_{\xi_i, \xi_i} = dE_{S\xi_i, T\xi_i} \ll \mu_i$. By (1) of \autoref{lemma:spectral-measure-properties} and completeness of $L^1(\mu; \complex)$, $\bracsn{E_{x, y}|x, y \in \ol{A\xi_i}}$ is absolutely continuous with respect to $\mu_i$. Therefore for any $C \in \cb_{\Omega(A)}$, $\mu_i(C) = 0$ if and only if $E_{x, y}(C) = 0$ for all $x, y \in \ol{A \xi_i}$. 
+
+    For any $i, j \in I$ with $i \ne j$, $x \in \ol{A\xi_i}$, $y \in \ol{A\xi_j}$, and $f \in C(\Omega(A); \complex)$, 
+    \[
+        \int_{\Omega(A)} f dE_{x, y} = \dpn{\Gamma_A^{-1}(f)x, y}{H} = 0
+    \]
+
+    because $\ol{A\xi_i} \perp \ol{A\xi_j}$, so $E_{x, y} = 0$. In particular, $\seqi{\mu}$ is a mutually singular family of measures, and $[l^1(I); L^1(\mu_i; \complex)]$ may be identified as a subspace of $M_R(\Omega(A); \complex)$. 
+
+    (2): For any $x, y \in H$, $\norm{E_{x, y}}_{\text{var}} \le \norm{x}_H \norm{y}_H$ by (1) of \autoref{lemma:spectral-measure-properties}. By (1), $\bracsn{E_{x, y}|x, y \in \ol{A\xi_i}} \subset [l^1(I); L^1(\mu_i; \complex)]$ for all $i \in I$. For each $i \in I$, let $P_i \in B(H)$ be the orthogonal projection of $H$ onto $\ol{A\xi_i}$, then as $\seqi{\xi}$ is maximal, for any $x = \sum_{i \in I}P_ix$ for all $x \in H$. Thus for any $x, y \in H$, 
+    \[
+        E_{x, y} = \sum_{i, j \in I}E_{P_ix, P_jy} =  \sum_{i \in I}E_{P_ix, P_iy} \in [l^1(I); L^1(\mu_i; \complex)]
+    \]
+    
+    so $\mathscr{E} \subset [l^1(I); L^1(\mu_i; \complex)]$. 
+
+    On the other hand, for each $i \in I$, since $\mu_i$ is a Radon measure, $C(\Omega(A); \complex)$ is dense in $L^1(\mu_i; \complex)$ by \autoref{proposition:radon-cc-dense}. As
+    \[
+    \mathscr{E} \supset \bracsn{E_{x, y}|x, y \in I} \supset \bracsn{fdE_{\xi_i, \xi_i}|f \in C(\Omega(A); \complex)}  = \bracsn{fd\mu_i|f \in C(\Omega(A); \complex)} 
+    \]
+
+    and $\mathscr{E} \subset M_R(\Omega(A); \complex)$ is closed, $\mathscr{E} \supset \bracsn{f d\mu_i|f \in L^1(\mu_i; \complex)}$. 
+    
+    Finally, given that the above holds for all $i \in I$, $\mathscr{E} = [l^1(I); L^1(\mu_i; \complex)]$. By \autoref{theorem:lp-sum-dual}, $[l^\infty(I); L^\infty(\mu_i; \complex)] = \mathscr{E}^*$. 
+    
+    (3): Fix $i \in I$, then for any $S, T \in A$ with $S\xi_i = T\xi_i$,
+    \[
+    \Gamma_AS dE_{\xi_i, \xi_i} = E_{\Gamma_A S\xi_i, \xi_i} = E_{\Gamma_A T\xi_i, \xi_i} = \Gamma_A T dE_{\xi_i, \xi_i}
+    \]
+
+    so $\Gamma_A S = \Gamma_A T$ $\mu_i$-almost everywhere. Thus the mapping
+    \[
+    U_i: \ol{A\xi_i} \to L^2(\mu_i; \complex) \quad T\xi_i \mapsto \Gamma_AT
+    \]
+
+    is well-defined. Moreover, for any $S, T \in A$, 
+    \[
+    \dpn{S\xi_i, T\xi_i}{H} = \int \Gamma_AS \cdot \ol{\Gamma_A T} dE_{\xi_i, \xi_i} = \dpn{\Gamma S, \Gamma T}{L^2(\mu_i; \complex)}
+    \]
+
+    so $U_i$ extends into an isometry between $\ol{A\xi_i}$ and $L^2(\mu_i; \complex)$. 
+
+    For each $i \in I$, let $P_i \in B(H)$ be the orthogonal projection of $H$ onto $\ol{A\xi_i}$, then
+    \[
+    U: H \to [l^2(I); L^2(\mu_i; \complex)] \quad (Ux)_i = U_i(P_ix)
+    \]
+
+    is an isometry between $H$ and $[l^2(I); L^2(\mu_i; \complex)]$ such that $U(Tx) = \Gamma_AT \cdot Ux$ for all $x \in H$ and $T \in A$. 
+    
+    Finally, given that
+    \begin{enumerate}[label=(\roman*)]
+        \item By \hyperref[Goldstine's Theorem]{corollary:weak-dense-unit-ball}, $C(X; \complex)$ is weak*-dense in $[l^\infty(I); L^\infty(\mu_i; \complex)]$. 
+        \item The weak* topology on $[l^\infty(I); L^\infty(\mu_i; \complex)]$ is equal to the weak operator topology of $[l^\infty(I); L^\infty(\mu_i; \complex)]$ acting on $[l^2(I); L^2(\mu_i; \complex)]$. 
+        \item $A$ is weak-operator dense in $B$. 
+        \item By \hyperref[Spectral Theorem I]{theorem:spectral-theorem-vn-1}, the isomorphism $\phi \mapsto \int \phi dE$ is continuous from the weak* topology on $\mathscr{E}^* = [l^\infty(I); L^\infty(\mu_i; \complex)]$ to the weak operator topology on $B$.
+    \end{enumerate}
+    
+    the mapping $U$ is an unitary equivalence between $[l^\infty(I); L^\infty(\mu_i; \complex)]$ acting on $[l^2(I); L^2(\mu_i; \complex)]$ and $B$ acting on $H$. 
+\end{proof}
+
+
+\begin{definition}[$L^\infty$ 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
     \begin{enumerate}