From f72f815c726695931066c3ae4e6aabd82fcb7c65 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Wed, 3 Jun 2026 15:44:13 -0400 Subject: [PATCH] Added basic facts about the Gelfand transform. --- src/op/banach/gelfand.tex | 60 ++++++++++++++++++++++++++++++++++ src/op/banach/index.tex | 1 + src/op/banach/spectrum.tex | 22 ++++++++++++- src/op/example/convolution.tex | 48 +++++++++++++++++++++++++++ src/op/example/index.tex | 1 + src/op/notation.tex | 4 +++ 6 files changed, 135 insertions(+), 1 deletion(-) create mode 100644 src/op/banach/gelfand.tex create mode 100644 src/op/example/convolution.tex diff --git a/src/op/banach/gelfand.tex b/src/op/banach/gelfand.tex new file mode 100644 index 0000000..f38d2c5 --- /dev/null +++ b/src/op/banach/gelfand.tex @@ -0,0 +1,60 @@ +\section{The Gelfand Transform} +\label{section:gelfand-transform} + +\begin{definition}[Gelfand Transform] +\label{definition:gelfand-transform} + Let $A$ be a unital Banach algebra, then the \textbf{Gelfand transform} is the homomorphism + \[ + \Gamma = \Gamma_A: A \to C(\Omega(A); \complex) \quad (\Gamma_Ax)(\varphi) = \varphi(x) + \] +\end{definition} + +\begin{remark} +\label{remark:gelfand-transform} + The Gelfand transform is limited in studying arbitrary Banach algebras, as they may admit no multiplicative functionals. However, these functionals come in abundance in the commutative case. +\end{remark} + + +\begin{proposition} +\label{proposition:gelfand-transform-gymnastics} + Let $A$ be a commutative unital Banach algebra and $x \in A$, then: + \begin{enumerate} + \item $\Gamma_A$ is a contractive homomorphism. + \item $\Gamma_A(1) = 1$. + \item $x \in G(A)$ if and only if $\Gamma_A x \in G(C(\Omega(A); \complex))$. + \item $(\Gamma_Ax)(\Omega(A)) = \sigma_A(x)$. + \item $\norm{\Gamma_Ax}_u = [x]_{sp}$. + \end{enumerate} +\end{proposition} +\begin{proof}[Proof, {{\cite[Theorem 1.1.13]{FollandHarmonic}}}. ] + (2): For each $\phi \in \Omega(A)$, $\phi(1) = 1$, so $\Gamma_A(1) = 1$. + + (3): Since $A$ is commutative, $x \not\in G(A)$ if and only if the ideal generated by $x$ is proper, if and only if there exists a maximal ideal containing $x$, if and only if there exists $\phi \in \Omega(A)$ with $\phi(x) = 0$. + + (4): By (1) and (3), + \[ + (\Gamma_Ax)(\Omega(A)) = \sigma_{C(\Omega(A); \complex)}(\Gamma x) = \sigma_A(x) + \] +\end{proof} + +\begin{proposition} +\label{proposition:gelfand-isometric} + Let $A$ be a commutative unital Banach algebra, then the following are equivalent: + \begin{enumerate} + \item For each $x \in A$, $\normn{x^2}_A = \norm{x}_A^2$. + \item $\Gamma_A$ is an isometry. + \end{enumerate} +\end{proposition} +\begin{proof} + (1) $\Rightarrow$ (2): For each $x \in A$, by the \hyperref[spectral radius formula]{proposition:spectral-radius-hadamard} and (5) of \autoref{proposition:gelfand-transform-gymnastics}, + \[ + \norm{\Gamma_A x}_u = [x]_{sp} = \norm{x}_A + \] + + (2) $\Rightarrow$ (1): For each $x \in A$, by (5) of \autoref{proposition:gelfand-transform-gymnastics}, + \[ + \normn{x^2}_A \ge [x^2]_{sp} = \normn{\Gamma_A x^2}_u = \normn{\Gamma_A x}_u^2 = \normn{x}_A^2 + \] +\end{proof} + + diff --git a/src/op/banach/index.tex b/src/op/banach/index.tex index aa071c7..5f4f746 100644 --- a/src/op/banach/index.tex +++ b/src/op/banach/index.tex @@ -8,3 +8,4 @@ \input{./spectrum.tex} \input{./fc.tex} \input{./multiplicative.tex} +\input{./gelfand.tex} diff --git a/src/op/banach/spectrum.tex b/src/op/banach/spectrum.tex index 40964ca..f5a9230 100644 --- a/src/op/banach/spectrum.tex +++ b/src/op/banach/spectrum.tex @@ -77,7 +77,7 @@ Let $x \in A$. By \autoref{proposition:spectrum-non-empty}, there exists $\lambda \in \sigma_A(x)$. Since $\lambda \cdot 1 - x$ is not invertible and every non-zero element of $A$ is invertible, $x = \lambda \cdot 1$. Therefore the mapping $\complex \to A$ defined by $\lambda \mapsto \lambda \cdot 1$ is an isometric isomorphism. \end{proof} -\begin{proposition} +\begin{proposition}[Spectral Radius Formula] \label{proposition:spectral-radius-hadamard} Let $A$ be a unital Banach algebra and $x \in A$, then $[\cdot]_{sp} = \limsup_{n \to \infty}\normn{x^n}_A^{1/n}$. \end{proposition} @@ -126,4 +126,24 @@ Let $x \in A$ with $\sigma_A(x) \subset U$, and $\lambda \in U^c$. By \autoref{proposition:banach-algebra-inverse}, for any $y \in A$ with $\norm{y}_A \le \normn{(\lambda - x)^{-1}}_A^{-1}$, $\lambda - x - y \in G(A)$ as well. Since the mapping $\lambda \mapsto \normn{(\lambda - x)^{-1}}_A$ vanishes at infinity and $U^c$ is closed, $\delta = \inf_{\lambda \in U^c}\normn{(\lambda - x)^{-1}}_A^{-1} > 0$. Therefore for every $z \in B_A(x,\delta)$, $\sigma_A(x) \subset U$. \end{proof} +\begin{proposition} +\label{proposition:commutative-spectrum-gymnastics} + Let $A$ be a unital Banach algebra and $x, y \in A$ with $x = y$, then + \begin{enumerate} + \item $\sigma_A(x + y) \subset \sigma_A(x) + \sigma_A(y)$. + \item $\sigma_A(xy) \subset \sigma_A(x)\sigma_A(y)$. + \end{enumerate} +\end{proposition} +\begin{proof} + For any $z \in A$, denote $R(z) = \bracs{(\lambda - z)^{-1}|\lambda \in \sigma_A(z)}$. Let $B \subset A$ be the closed subalgebra generated by $1$, $x$, $y$, $R(x)$, $R(y)$, $R(xy)$, $R(x + y)$, then $B$ is a commutative algebra with $\sigma_B(x) = \sigma_A(x)$, $\sigma_B(y) = \sigma_A(y)$, $\sigma_B(xy) = \sigma_A(xy)$, and $\sigma_B(x + y) = \sigma_A(x + y)$. + + By \autoref{proposition:gelfand-transform-gymnastics}, for each $z \in B$, $(\Gamma_Bz)(\Omega(B)) = \sigma_B(z)$. Since for any $u, v \in C(\Omega(B); \complex)$, + \begin{enumerate} + \item $\sigma_{C(\Omega(B); \complex)}(u + v) \subset \sigma_{C(\Omega(B); \complex)}(u) + \sigma_{C(\Omega(B); \complex)}(v)$. + \item $\sigma_{C(\Omega(B); \complex)}(uv) \subset \sigma_{C(\Omega(B); \complex)}(u)\sigma_{C(\Omega(B); \complex)}(v)$. + \end{enumerate} + + The above holds for $x$ and $y$ with respect to $\sigma_A$. +\end{proof} + diff --git a/src/op/example/convolution.tex b/src/op/example/convolution.tex new file mode 100644 index 0000000..7827676 --- /dev/null +++ b/src/op/example/convolution.tex @@ -0,0 +1,48 @@ +\section{$\ell^1(\integer)$} +\label{section:convolution-algebra-integer} + +\begin{definition}[$\ell^1(\integer)$] +\label{definition:convolution-algebra-integer} + Let $\ell^1(\integer)$ be the $\ell^1$ sequence space on $\integer$. For each $f, g \in \ell^1(\integer)$, let + \[ + (f * g)(n) = \sum_{k \in \integer}f(k)g(n - k) + \] + + then: + \begin{enumerate} + \item $\ell^1(\integer)$ is a commutative Banach algebra. + \item The multiplicative unit of $\ell^1(\integer)$ is $\delta_0 = \one_{\bracs{n = 0}}$. + \end{enumerate} + + The space $\ell^1(\integer)$ is the \textbf{convolution algebra} on $\integer$. +\end{definition} +\begin{proof} + For each $f, g \in \ell^1(\integer)$, + \begin{align*} + \normn{f * g}_{\ell^1(\integer)} &= \sum_{n \in \integer} \abs{\sum_{k \in \integer}f(k)g(n - k)} \\ + &\le \sum_{n, k \in \integer}|f(k)| \cdot |g(n-k)| \le \sum_{k \in \integer}|f(k)| \cdot \sum_{n \in \integer}|g(n - k)| \\ + &= \norm{f}_{\ell^1(\integer)} \cdot \norm{g}_{\ell^1(\integer)} + \end{align*} +\end{proof} + + +\begin{proposition} +\label{proposition:convolution-integer-gelfand} + The Gelfand transform of $\ell^1(\integer)$ is not isometric. +\end{proposition} +\begin{proof} + Let $f = \one_{\bracs{n = 1}} - \one_{\bracs{2 \le n \le 3}}$, then + \[ + f^2(n) = \begin{cases} + -1 &n \in \bracs{1, 5} \\ + -2 &n = 2 \\ + -1 &n = 3 \\ + 2 &n = 4 \\ + 0 &n \not\in [1, 5] + \end{cases} + \] + + so $\normn{f^2}_{\ell^1(\integer)} = 7 < \normn{f}_{\ell^1(\integer)}^2$. By \autoref{proposition:gelfand-isometric}, the Gelfand transform is not isometric. +\end{proof} + + diff --git a/src/op/example/index.tex b/src/op/example/index.tex index 4077be3..6614fc7 100644 --- a/src/op/example/index.tex +++ b/src/op/example/index.tex @@ -5,4 +5,5 @@ \input{./bounded.tex} \input{./hardy.tex} \input{./disk.tex} +\input{./convolution.tex} diff --git a/src/op/notation.tex b/src/op/notation.tex index 6db3ce1..23aec77 100644 --- a/src/op/notation.tex +++ b/src/op/notation.tex @@ -13,8 +13,12 @@ $[x]_{sp}$ & The spectral radius of $x$. & \autoref{definition:spectral-radius} \\ $\Omega(A)$ & 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} \\ + $M_n(\complex)$ & 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} \\ $A(D)$ & The disk algebra. & \autoref{definition:disk-algebra} \\ $H^\infty(D)$ & The Hardy space. & \autoref{definition:hardy-space} \\ + $\ell^1(\integer)$ & Convolution algebra on $\integer$. &\autoref{definition:convolution-algebra-integer} \\ + $\delta_0$ & Multiplicative unit of $\ell^1(\integer)$. & \autoref{definition:convolution-algebra-integer} \\ \end{tabular} \ No newline at end of file