From 13a0e07b72a54eb63d10ea84be71ab92fae50644 Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Thu, 4 Jun 2026 13:41:05 -0400 Subject: [PATCH] Fixed some numbering problems. --- src/op/banach/gelfand.tex | 12 ++++++------ src/op/example/convolution.tex | 4 ++-- 2 files changed, 8 insertions(+), 8 deletions(-) diff --git a/src/op/banach/gelfand.tex b/src/op/banach/gelfand.tex index 5345bba..8353bb7 100644 --- a/src/op/banach/gelfand.tex +++ b/src/op/banach/gelfand.tex @@ -3,7 +3,7 @@ \begin{definition}[Gelfand Transform] \label{definition:gelfand-transform} - Let $A$ be a unital Banach algebra, then the \textbf{Gelfand transform} is the homomorphism + Let $A$ be a unital Banach algebra, then the \textbf{Gelfand transform} is the contractive homomorphism \[ \Gamma = \Gamma_A: A \to C(\Omega(A); \complex) \quad (\Gamma_Ax)(\varphi) = \varphi(x) \] @@ -26,11 +26,11 @@ \end{enumerate} \end{proposition} \begin{proof}[Proof, {{\cite[Theorem 1.1.13]{FollandHarmonic}}}. ] - (2): For each $\phi \in \Omega(A)$, $\phi(1) = 1$, so $\Gamma_A(1) = 1$. + (1): 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 $\phi(x) = 0$. + (2): 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), + (3): By (2), \[ (\Gamma_Ax)(\Omega(A)) = \sigma_{C(\Omega(A); \complex)}(\Gamma x) = \sigma_A(x) \] @@ -45,12 +45,12 @@ \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}, + (1) $\Rightarrow$ (2): For each $x \in A$, by the \hyperref[spectral radius formula]{proposition:spectral-radius-hadamard} and (4) 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}, + (2) $\Rightarrow$ (1): For each $x \in A$, by (4) of \autoref{proposition:gelfand-transform-gymnastics}, \[ \normn{x^2}_A \ge [x^2]_{sp} = \normn{\Gamma_A x^2}_u = \normn{\Gamma_A x}_u^2 = \normn{x}_A^2 \] diff --git a/src/op/example/convolution.tex b/src/op/example/convolution.tex index b315d58..c259efd 100644 --- a/src/op/example/convolution.tex +++ b/src/op/example/convolution.tex @@ -77,11 +77,11 @@ \label{proposition:convolution-integer-spectrum} Let $\ell^1(\integer)$ be the convolution algebra on $\integer$ and $f \in \ell^1(\integer)$, then \[ - \sigma(f) = \bracs{\sum_{n \in \integer}f(n)z^n \bigg | z \in \partial B_\complex(0, 1)} + \sigma_{\ell^1(\integer)}(f) = \bracs{\sum_{n \in \integer}f(n)z^n \bigg | z \in \partial B_\complex(0, 1)} \] \end{proposition} \begin{proof} - By \autoref{theorem:convolution-integer-gelfand} and (4) of \autoref{proposition:gelfand-transform-gymnastics}. + By \autoref{theorem:convolution-integer-gelfand} and (3) of \autoref{proposition:gelfand-transform-gymnastics}. \end{proof}