Added first draft of the Borel functional calculus.

src/fa/lp/definition.tex

@@ -1,6 +1,12 @@
 \section{Basic Properties}
 \label{section:lp-basic}
 
+\begin{definition}[$B^\infty$ Spaces]
+\label{definition:bounded-borel-function}
+    Let $(X, \cm, \mu)$ be a measure space and $E$ be a normed vector space, then the set $B^\infty(X; E)$ is the \textbf{space of bounded $E$-valued strongly measurable functions} on $X$, and the set $B^\infty(X)$ is the space of bounded complex-valued Borel measurable functions on $X$. 
+\end{definition}
+
+
 \begin{definition}[$\mathcal{L}^p$ Spaces]
 \label{definition:lp-unequivalence}
     Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, $f: X \to E$ be strongly measurable, and $p \in [1, \infty)$, then $f$ is \textbf{$p$-integrable} if
@@ -8,17 +14,19 @@
     \norm{f}_{L^p(X; E)} = \norm{f}_{L^p(\mu; E)} = \norm{f}_{L^p(X, \cm, \mu; E)} = \braks{\int \norm{f}_E^p d\mu}^{1/p} < \infty
     \]
 
-    The set $\mathcal{L}^p(X; E) = \mathcal{L}^p(\mu; E) = \mathcal{L}^p(X, \cm, \mu; E)$ is the space of all $p$-integrable functions on $X$.
+    The set $\mathcal{L}^p(X; E) = \mathcal{L}^p(\mu; E) = \mathcal{L}^p(X, \cm, \mu; E)$ is the space of all $E$-valued $p$-integrable functions on $X$.
 \end{definition}
 
 \begin{definition}[Essential Supremum]
 \label{definition:esssup}
     Let $(X, \cm, \mu)$ be a measure space, $E$ be a normed vector space, and $f: X \to E$ be strongly measurable, then $f$ is \textbf{essentially bounded} if
     \[
-    \norm{f}_{L^\infty(X; E)} = \norm{f}_{L^\infty(\mu; E)} = \norm{f}_{L^\infty(X, \cm, \mu; E)} = \inf\bracs{\alpha \ge 0|\mu(\bracs{f > \alpha}) = 0} < \infty
+    \norm{f}_{\mathcal{L}^\infty(X; E)} = \norm{f}_{\mathcal{L}^\infty(\mu; E)} = \norm{f}_{\mathcal{L}^\infty(X, \cm, \mu; E)} = \inf\bracs{\alpha \ge 0|\mu(\bracs{f > \alpha}) = 0} < \infty
     \]
 
-    In which case, $\norm{f}_{L^\infty(X; E)}$ is the \textbf{essential supremum} of $f$.
+    In which case, $\norm{f}_{\mathcal{L}^\infty(X; E)}$ is the \textbf{essential supremum} of $f$. 
+    
+    The set $\mathcal{L}^\infty(X; E) = \mathcal{L}^\infty(\mu; E) = \mathcal{L}^\infty(X, \cm, \mu; E)$ is the space of all $E$-valued essentially bounded functions on $X$. 
 \end{definition}
 
 \begin{definition}[Hölder conjugates]

src/op/notation.tex

@@ -20,6 +20,7 @@
     $\dpn{x, y}{\phi}$ & Defined as $\dpn{y^*x, \phi}{A}$, the pseudo inner product associated to a positive linear functional. & \autoref{definition:cstar-state-pseudo-inner-product} \\ 
     $(H_\phi, \pi_\phi, \xi_\phi)$ & GNS triple associated with $\phi \in S(A)$. & \autoref{definition:gns-triple} \\ 
     $U(T)$ & Cayley transform of $T$. & \autoref{definition:cayley-transform-bounded} \\ 
+    $E_{x, y}$ & $E_{x, y}(B) = \dpn{E(B)x, y}{H}$. & \autoref{definition:spectr}
 
     $M_n(\complex)$ & Algebra of $n \times n$ matrices over $\complex$.  & \autoref{definition:matrix-algebra} \\
     $B(H)$ & Algebra of bounded operators on a Hilbert space. & \autoref{definition:hilbert-endomorphism} \\

src/op/vn/fc.tex

@@ -1,19 +1,181 @@
 \section{The Borel Functional Calculus}
 \label{section:borel-functional-calculus}
 
+\begin{definition}[Spectral Measure]
+\label{definition:spectral-measure}
+    Let $X$ be a compact Hausdorff space, $H$ be a complex Hilbert space, and $E: \cb_X \to B(H)$, then $E$ is a \textbf{spectral measure relative to $H$} if:
+    \begin{enumerate}
+        \item For each $B \in \cb_X$, $E(B)$ is an orthogonal projection. 
+        \item $E(\emptyset) = 0$, $E(X) = I_{B(H)}$. 
+        \item For each $B, C \in \cb_X$, $E(B \cap C) = E(B)E(C)$. 
+        \item For each $x, y \in H$, the mapping
+        \[
+        E_{x, y}: \cb_X \to \complex \quad B \mapsto \dpn{E(B)x, y}{H}
+        \]
+
+        is a complex Radon measure on $X$. 
+    \end{enumerate}
+    
+\end{definition}
+
+\begin{definition}[Integration Against a Spectral Measure]
+\label{definition:spectral-measure-integral}
+    Let $X$ be a compact Hausdorff space, $H$ be a complex Hilbert space, and $E: \cb_X \to B(H)$ be a spectral measure. Define
+    \[
+    I_E: C(X; \complex)^{**} \to B(H) \quad \phi \mapsto \int_X \phi dE
+    \]
+
+    where for each $x, y \in H$, $\angles{I_E(\phi) \cdot x, y}_{H} = \dpn{E_{x, y}, \phi}{C(X; \complex)^*}$, then:
+    \begin{enumerate}
+        \item $I_E$ is continuous from the weak*-topology on $C(X; \complex)^{**}$ to the weak operator topology on $B(H)$. 
+        \item $I_E$ is a unital *-homomorphism. 
+    \end{enumerate}
+
+    For any $\phi \in C(X; \complex)^{**}$, $I_E(\phi) = \int_X \phi dE$ is the \textbf{integral} of $\phi$ with respect to $E$. 
+\end{definition}
+\begin{proof}
+    Firstly, let $x, y \in H$, $\seqf{B_j} \subset \cb_X$ be disjoint Borel sets, and $B = \bigsqcup_{j = 1}^n B_j$, then for each $1 \le i < j \le n$, $E(B_i)(H) \perp E(B_j)(H)$, so by the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz} and the \hyperref[Pythagorean Theorem]{theorem:pythagoras},
+    \begin{align*}
+        \sum_{j = 1}^n |\dpn{E(B_j)x, y}{H}| &= \sum_{j = 1}^n |\dpn{E(B_j)x, E(B_j)y}{H}| \\
+        &\le \sum_{j = 1}^n \norm{E(B_j)x}_H \norm{E(B_j)y}_H \\
+        &\le \braks{\sum_{j = 1}^n \norm{E(B_j)x}_H^2}^{1/2} \cdot \braks{\sum_{j = 1}^n \norm{E(B_j)y}_H^2}^{1/2} \\
+        &= \norm{E(B)x}_H \cdot \norm{E(B)y}_H \le \norm{x}_H \cdot \norm{y}_H
+    \end{align*}
+
+    As the above holds for all finite sequences of disjoint Borel sets, $\norm{E_{x, y}}_{C(X; \complex)^*} \le \norm{x}_H \norm{y}_H$. Thus for any $\phi \in C(X; \complex)^{**}$, 
+    \begin{align*}
+    |\dpn{I_E(\phi) \cdot x, y}{H}| &= |\dpn{E_{x, y}, \phi}{C(X; \complex)^*}| \le \norm{E_{x, y}}_{C(X; \complex)^*} \cdot \norm{\phi}_{C(X; \complex)^{**}} \\
+    &\le \norm{\phi}_{C(X; \complex)^{**}} \cdot \norm{x}_H \cdot \norm{y}_H
+    \end{align*}
+
+    Since the above holds for all $x, y \in H$, $I_E(\phi) \in B(H)$ with $\norm{I_E(\phi)}_{B(H)} \le \norm{\phi}_{C(X; \complex)^{**}}$. 
+    
+
+    (1): For each $x, y \in H$, $E_{x, y} \in C(X; \complex)^*$. Since $\angles{\int \phi dE \cdot x, y}_{H} = \dpn{E_{x, y}, \phi}{C(X; \complex)^*}$ for every $\phi \in C(X; \complex)^{**}$, $I_E$ is continuous from the weak* topology on $C(X; \complex)^{**}$ to the weak operator topology on $B(H)$. 
+
+    (2): By \autoref{lemma:separable-metric-space-approx-identity}, the simple functions $\Sigma(X; \complex)$ are uniformly dense in the bounded Borel functions $B^\infty(X; \complex)$. Since
+    \begin{enumerate}[label=(\roman*)]
+        \item $I_E$ restricted to $\Sigma(X; \complex)$ is a *-homomorphism. 
+        \item Multiplication and conjugation are continuous in the uniform norm on $B^\infty(X; \complex)$
+        \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]{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 $C(X; \complex)^{**}$. 
+        \item The transpose $T \mapsto T^*$ is weak-operator continuous on $B(H)$. 
+        \item The product $(\phi, \psi) \mapsto \phi \psi$ is separately weak*-continuous on $C(X; \complex)^{**}$. 
+        \item The composition $(S, T) \mapsto ST$ is separately weak-operator continuous on $B(H)$.
+    \end{enumerate}
+
+    the map $I_E$ is a *-homomorphism by (1). Finally, since $E(X) = I_{B(H)}$, $I_E$ is a unital *-homomorphism. 
+\end{proof}
+
+\begin{theorem}[Spectral Theorem (I)]
+\label{theorem:spectral-theorem-vn-1}
+    Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a commutative $C^*$-subalgebra with $I \in A$, then:
+    \begin{enumerate}
+        \item There exists a unique spectral measure $E: \cb_{\Omega(A)} \to B(H)$ such that
+        \[
+        T = \int_{\Omega(A)} \Gamma_A T dE \quad \forall T \in A
+        \]
+        \item Let $B \subset B(H)$ be the strong-operator closure of $A$ in $B(H)$, then
+        \[
+        B = \bracs{\int_{\Omega(A)} \phi dE \bigg | \phi \in C(\Omega(A); \complex)^{**}}
+        \]
+    \end{enumerate}
+    
+    The mapping $C(\Omega(A); \complex)^{**} \to B$ defined by $\phi \mapsto \int \phi dE$ 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 unital *-isomorphism. For each $x, y \in H$, $\Gamma_A^{-1}$ induces a mapping
+    \[
+    E_{x, y}: C(\Omega(A); \complex) \to \complex \quad \dpn{f, E_{x, y}}{C(\Omega(A); \complex)} = \dpn{\Gamma_A^{-1}f \cdot x, y}{H}
+    \]
+
+    which, by the \hyperref[Riesz Representation Theorem]{theorem:riesz-radon-c0}, takes the form of a complex Radon measure on $\Omega(A)$. Thus by the uniqueness part of the Riesz Representation Theorem, such a spectral measure must be unique if it exists. 
+
+    Since $\norm{E_{x, y}}_{C(\Omega(A); \complex)^*} \le \norm{x}_H\norm{y}_H$ for all $x, y \in H$, $\bracsn{E_{x, y}|x, y \in H}$ induces a bounded linear map
+    \[
+    I: C(\Omega(A); \complex)^{**} \to B(H) \quad \dpn{I(\phi)x, y}{H} = \dpn{E_{x, y}, \phi}{C(\Omega(A); \complex)^*}
+    \]
+
+    with $I(f) = \Gamma_A^{-1}(f)$ for all $f \in C(\Omega(A); \complex)$. 
+
+    For any $C \in \cb_{\Omega(A)}$, $\one_C$ is a projection in $B^\infty(\Omega(A); \complex)$. So to see that
+    \[
+    E: \cb_{\Omega(A)} \to B(H) \quad \dpn{E(C)x, y}{H} = E_{x, y}(C)
+    \]
+
+    defines a spectral measure, it is sufficient to show that $I|_{B^\infty(\Omega(A); \complex)}$ is a *-homomorphism. 
+
+    Let $x, y \in H$, then as $\Gamma_A$ is a *-isomorphism, for any $f, g \in C(\Omega(A); \complex)$, 
+    \begin{align*}
+        \dpn{fg, E_{x, y}}{C(\Omega(A); \complex)} &= \dpn{\Gamma_A^{-1}f \cdot \Gamma_A^{-1}g \cdot x, y}{H} \\
+        &= \dpn{\Gamma_A^{-1}g \cdot x, (\Gamma_A^{-1}f)^* y}{H} = \dpn{g, E_{x, I(f)^*y}}{C(\Omega(A); \complex)}
+    \end{align*}
+
+    As the above holds for all $g \in C(\Omega(A); \complex)$, $fE_{x, y} = E_{x, I(f)^*y}$. Now, fix $\phi \in B^\infty(\Omega(A); \complex)$, then for every $f \in C(\Omega(A); \complex)$, 
+    \begin{align*}
+        \dpn{E_{x, y}, \phi f}{C(\Omega(A); \complex)^*} &= \dpn{E_{x, I(f)^*y}, \phi}{C(\Omega(A); \complex)^*} = \dpn{I(\phi)x, I(f)^*y}{H} \\
+        &= \dpn{I(f)I(\phi)x, y}{H} = \dpn{f, E_{I(\phi)x, y}}{C(\Omega(A); \complex)}
+    \end{align*}
+
+    so $\phi E_{x, y} = E_{I(\phi)x, y}$ for all $\phi \in B^\infty(\Omega(A); \complex)$. Thus for any $\phi, \psi \in B^\infty(\Omega(A); \complex)$, 
+    \begin{align*}
+    \dpn{I(\phi \psi)x, y}{H} &= \dpn{E_{x, y}, \phi \psi}{C(\Omega(A); \complex)^*} = \dpn{E_{I(\psi) x, y}, \phi}{C(\Omega(A); \complex)^*} \\
+    &= \dpn{I(\phi)I(\psi)x, y}{H}
+    \end{align*}
+
+    and $I|_{B^\infty(\Omega(A); \complex)}$ is a homomorphism\footnote{With the same amount of writing and considerably more mental gymnastics, it can be shown that $I$ is a *-homomorphism on the full space $C(\Omega(A); \complex)^{**}$. However, it is not needed to show that $E$ is a spectral measure, and the homomorphism property falls out at the end anyways.}. 
+
+    Finally, let $f \in C(\Omega(A); \real)$, then since $\Gamma_A$ is a *-isomorphism, $I(f) = \Gamma_A^{-1}(f)$ is self-adjoint. As such, for any $x \in H$, $\dpn{f, E_{x, x}}{C(\Omega(A); \complex)} = \dpn{I(f)x, x}{H} \in \real$, so $E_{x, x}$ is real-valued. Thus for any $\phi \in B^\infty(\Omega(A); \real)$ and $x \in H$, $\dpn{I(\phi)x, x}{H} = \dpn{E_{x, x}, \phi}{C(\Omega(A); \complex)^*} \in \real$ as well. Therefore $I(\phi)$ is self-adjoint, and $I|_{B^\infty(\Omega(A); \complex)}$ is a *-homomorphism. 
+
+    (2): By \hyperref[Goldstine's Theorem]{corollary:weak-dense-unit-ball}, $C(\Omega(A); \complex)$ is weak*-dense in $C(\Omega(A); \complex)^{**}$. Since the mapping $\phi \mapsto \int_{\Omega(A)}\phi dE$ is continuous from the weak* topology on $C(\Omega(A); \complex)^{**}$ to the weak operator topology on $B(H)$, 
+    \[
+    B \supset \bracs{\int_{\Omega(A)} \phi dE \bigg | \phi \in C(\Omega(A); \complex)^{**}}
+    \]
+
+    by \autoref{proposition:closure-of-image}. 
+    
+    On the other hand, by the \hyperref[Banach-Alaoglu Theorem]{theorem:alaoglu}, $\ol{B_{C(\Omega(A); \complex)^{**}}(0, 1)}$ is weak*-compact, so
+    \[
+    S := \bracs{\int_{\Omega(A)} \phi dE \bigg | \phi \in C(\Omega(A); \complex)^{**}, \norm{\phi}_{C(\Omega(A); \complex)^{**}} \le 1}
+    \]
+
+    is weak-operator compact. As $\Gamma_A: A \to C(\Omega(A); \complex)$ is an isometric isomorphism, $S \supset \ol{B_A(0, 1)}$. By the \hyperref[Kaplansky Density Theorem]{theorem:kaplansky-density}, $S \supset B_B(0, 1)$, and
+    \[
+    B = \bracs{\int_{\Omega(A)} \phi dE \bigg | \phi \in C(\Omega(A); \complex)^{**}}
+    \]
+\end{proof}
+
 \begin{definition}[Borel Functional Calculus]
 \label{definition:borel-functional-calculus}
-    Let $H$ be a complex Hilbert space, $A \subset B(H)$ be a von Neumann algebra, and $x \in A$ be normal, then there exists a unique continuous unital *-homomorphism
+    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
     \[
-    C(\sigma_A(x); \complex)^{**} \to A[x] \quad f \mapsto f(x)
+    C(\sigma_{B(H)}(T); \complex)^{**} \to A \quad \phi \mapsto \phi(T)
     \]
 
     such that:
     \begin{enumerate}
-        \item $\one(x) = 1_A$, $\text{Id}(x) = x$, and $\overline{\text{Id}}(x) = x^*$.
-        \item The mapping $f \mapsto f(x)$ is continuous from the weak* topology on $C(\sigma_A(x); \complex)^{**}$ to the strong operator topology on $B(H)$. 
+        \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}
-    Since the \autoref{definition:continuous-functional-calculus}
+    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}
+
+

src/op/vn/index.tex

@@ -3,4 +3,5 @@
 
 \input{./topologies.tex}
 \input{./cayley.tex}
-\input{./vn.tex}
+\input{./vn.tex}
+\input{./fc.tex}

src/topology/notation.tex

@@ -26,6 +26,14 @@
     $f \prec U$ & $f \in C_c(X; [0,1])$ with $\mathrm{supp}(f) \subset U$. & \autoref{definition:compactly-supported-01} \\
     $C_0(X; E)$ & Continuous functions vanishing at infinity. & \autoref{definition:vanish-at-infinity} \\
     $BC(X; E)$ & Bounded continuous functions $X \to E$. & \autoref{definition:bounded-continuous-function-space} \\
+    % ---- $L^p$ Spaces ----
+    $B^\infty(X; E)$ & Bounded $E$-valued strongly measurable functions on $X$. & \autoref{definition:bounded-borel-function} \\
+    $B^\infty(X)$ & Bounded $\complex$-valued Borel measurable functions on $X$. & \autoref{definition:bounded-borel-function} \\
+    $\mathcal{L}^p(X; E)$, $\mathcal{L}^p(\mu; E)$, $\mathcal{L}^p(X, \cm, \mu; E)$ & $E$-valued $p$-integrable functions on $X$. & \autoref{definition:lp-unequivalence} \\
+    $\norm{f}_{L^p(X; E)}$ & $L^p$ norm of $f$: $\braks{\int \norm{f}_E^p d\mu}^{1/p}$. & \autoref{definition:lp-unequivalence} \\
+    $\mathcal{L}^\infty(X; E)$, $\mathcal{L}^\infty(\mu; E)$, $\mathcal{L}^\infty(X, \cm, \mu; E)$ & $E$-valued essentially bounded functions on $X$. & \autoref{definition:esssup} \\
+    $\norm{f}_{\mathcal{L}^\infty(X; E)}$ & Essential supremum of $f$. & \autoref{definition:esssup} \\
+    $L^p(X, \cm, \mu; E)$ & $E$-valued $L^p$ space on $(X,\cm,\mu)$; quotient of $\mathcal{L}^p$ by a.e.-equality. & \autoref{definition:lp} \\
 
     % DST 
     $\mathscr{N}$ & The Baire space. & \autoref{definition:the-baire-space} \\