diff --git a/src/op/notation.tex b/src/op/notation.tex index 67bf03c..ac23d3c 100644 --- a/src/op/notation.tex +++ b/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} \\ diff --git a/src/op/vn/fc.tex b/src/op/vn/fc.tex index 7b97a2b..380ae98 100644 --- a/src/op/vn/fc.tex +++ b/src/op/vn/fc.tex @@ -1,19 +1,151 @@ \section{The Borel Functional Calculus} \label{section:borel-functional-calculus} -\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 +\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 \[ - C(\sigma_A(x); \complex)^{**} \to A[x] \quad f \mapsto f(x) + I_E: C(X; \complex)^{**} \to B(H) \quad \phi \mapsto \int_X \phi dE \] - such that: + 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 $\one(x) = 1_A$, $\text{Id}(x) = x$, 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 $I_E$ 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} - Since the \autoref{definition:continuous-functional-calculus} + 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]{theorem:goldstine-weak}, $C(X; \complex) \subset B^\infty(X; \complex)$ 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} \ No newline at end of file diff --git a/src/op/vn/index.tex b/src/op/vn/index.tex index 9280a14..72555d3 100644 --- a/src/op/vn/index.tex +++ b/src/op/vn/index.tex @@ -3,4 +3,5 @@ \input{./topologies.tex} \input{./cayley.tex} -\input{./vn.tex} \ No newline at end of file +\input{./vn.tex} +\input{./fc.tex} \ No newline at end of file