From 35e9550ff2886ed86ee70a8c27dce45f35a4c9bf Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Fri, 3 Jul 2026 15:21:33 -0400 Subject: [PATCH] Added the continuous functional calculus. --- src/op/banach/invertible.tex | 11 ++++-- src/op/banach/spectrum.tex | 25 ++++++++++++++ src/op/c-star/cont.tex | 65 ++++++++++++++++++++++++++++++++++++ src/op/c-star/index.tex | 4 ++- src/op/c-star/involution.tex | 13 -------- src/op/c-star/sub.tex | 41 +++++++++++++++++++++++ src/op/notation.tex | 1 + 7 files changed, 144 insertions(+), 16 deletions(-) create mode 100644 src/op/c-star/cont.tex create mode 100644 src/op/c-star/sub.tex diff --git a/src/op/banach/invertible.tex b/src/op/banach/invertible.tex index 512b51b..320de3a 100644 --- a/src/op/banach/invertible.tex +++ b/src/op/banach/invertible.tex @@ -52,6 +52,15 @@ (3): Since the inversion map is locally a power series, it is $C^\infty$ by \autoref{theorem:termwise-differentiation}. \end{proof} +\begin{corollary} +\label{corollary:invertible-boundary-explode} + Let $A$ be a unital Banach algebra, $x \in A \setminus G(A)$, and $r > 0$, then $\normn{y^{-1}}_A > 1/r$ for all $y \in B(x, r) \cap G(A)$. +\end{corollary} +\begin{proof} + Let $y \in B(x, r)$, then $B(y, \normn{y^{-1}}_A^{-1}) \subset G(A)$ by \autoref{proposition:banach-algebra-inverse}. Since $x \not\in G(A)$, $r > \norm{x - y}_A \ge \normn{y^{-1}}_A^{-1}$, and $1/r < \normn{y^{-1}}_A$. +\end{proof} + + \begin{proposition} \label{proposition:swap-invertible} Let $A$ be a unital Banach algebra and $x, y \in A$, then $1 - xy \in G(A)$ if and only if $1 - yx \in G(A)$. @@ -74,5 +83,3 @@ \end{align*} \end{proof} - - diff --git a/src/op/banach/spectrum.tex b/src/op/banach/spectrum.tex index f47fcc8..4559c2d 100644 --- a/src/op/banach/spectrum.tex +++ b/src/op/banach/spectrum.tex @@ -146,4 +146,29 @@ The above holds for $x$ and $y$ with respect to $\sigma_A$. \end{proof} +\begin{proposition} +\label{proposition:spectrum-subalgebra-gymnastics} + Let $A$ be a unital Banach algebra, $B \subset A$ be a closed subalgebra containing $1$, and $x \in B$, then: + \begin{enumerate} + \item $\sigma_A(x) \subset \sigma_B(x)$. + \item $\partial \sigma_B(x) \subset \sigma_A(x)$. + \item $\sigma_B(x)$ is the union of $\sigma_A(x)$ and some bounded components of $\complex \setminus \sigma_A(x)$. + \end{enumerate} +\end{proposition} +\begin{proof} + (1): $G(B) \subset G(A)$. + + (2): Let $\lambda \in \partial \sigma_B(x)$, then there exists $\seq{\lambda_n} \subset \complex \setminus \sigma_B(x)$ such that $\lambda_n - x \in G(B)$ for all $n \in \natp$, and $\lambda_n \to \lambda$ as $n \to \infty$. By \autoref{corollary:invertible-boundary-explode}, $\norm{(\lambda_n - x)^{-1}}_A \to \infty$ as $n \to \infty$. If $\lambda - x \in G(A)$, then $(\lambda_n - x)^{-1} \to (\lambda - x)^{-1}$ as $n \to \infty$. Thus $\norm{(\lambda - x)^{-1}}_A = \infty$, which is impossible. Therefore $\lambda - x \not\in G(A)$, and $\lambda \in \sigma_A(x)$. +\end{proof} + +\begin{theorem}["Runge's Theorem"] +\label{theorem:spectrum-subalgebra-sufficiency} + Let $A$ be a unital Banach algebra, $x \in A$, $P \subset \complex \setminus \sigma_A(x)$ such that $P$ intersects every bounded component of $\complex \setminus \sigma_A(x)$, and $B \subset A$ be a closed algebra containing $1$, $x$, and $\bracsn{(\lambda - x)^{-1}|\lambda \in P}$, then $\sigma_A(x) = \sigma_B(x)$. +\end{theorem} +\begin{proof}[Proof, {{\cite[Theorem 4.9]{MarcouxNotes}}}. ] + By construction, $P \subset \complex \setminus \sigma_B(x)$. In addition, for any polynomial $p \in \complex[z]$, $p(x) \in B$. Thus for every rational function $f \in \complex(z) \cap H(\complex_\infty \setminus (P \cup \bracs{\infty}); \complex)$, $f(x) \in B$. + + By \hyperref[Runge's Theorem]{theorem:runge}, $H(\complex_\infty \setminus (P \cup \bracs{\infty}); \complex)$ is dense in $H(\sigma_A(x); \complex)$. The continuity of the \hyperref[holomorphic functional calculus]{definition:holomorphic-functional-calculus} then implies that $f(x) \in B$ for all $f \in H(\sigma_A(x); \complex)$. In particular, $(\lambda - x)^{-1} \in B$ for all $\lambda \in \complex \setminus \sigma_A(x)$. Therefore $\sigma_B(x) \subset \sigma_A(x)$, and $\sigma_B(x) = \sigma_A(x)$ by \autoref{proposition:spectrum-subalgebra-gymnastics}. +\end{proof} + diff --git a/src/op/c-star/cont.tex b/src/op/c-star/cont.tex new file mode 100644 index 0000000..184b06b --- /dev/null +++ b/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} + + + diff --git a/src/op/c-star/index.tex b/src/op/c-star/index.tex index 6e4ceb8..bf7240d 100644 --- a/src/op/c-star/index.tex +++ b/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} \ No newline at end of file +\input{./gelfand.tex} +\input{./cont.tex} \ No newline at end of file diff --git a/src/op/c-star/involution.tex b/src/op/c-star/involution.tex index 83219c0..3b3986a 100644 --- a/src/op/c-star/involution.tex +++ b/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} - - - diff --git a/src/op/c-star/sub.tex b/src/op/c-star/sub.tex new file mode 100644 index 0000000..2103da0 --- /dev/null +++ b/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} + + diff --git a/src/op/notation.tex b/src/op/notation.tex index 23aec77..9e6914c 100644 --- a/src/op/notation.tex +++ b/src/op/notation.tex @@ -14,6 +14,7 @@ $\Omega(A)$ & Space of multiplicative functionals on $A$. & \autoref{definition:multiplicative-functional} \\ $\cm(A)$ & Maximal ideal space of $A$. & \autoref{definition:maximal-ideal} \\ $\Gamma = \Gamma_A$ & The Gelfand transform on $A$. & \autoref{definition:gelfand-transform} \\ + $A[S]$ & $C^*$-subalgebra of $A$ generated by $S \subset A$. & \autoref{definition:generated-subalgebra} \\ $M_n(\complex)$ & 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} \\