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Fixed some numbering problems.

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This commit is contained in:
Bokuan Li
2026-06-04 13:41:05 -04:00
parent ebead6c022
commit 13a0e07b72
2 changed files with 8 additions and 8 deletions
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12
src/op/banach/gelfand.tex
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@@ -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},
\[
\normn{x^2}_A \ge [x^2]_{sp} = \normn{\Gamma_A x^2}_u = \normn{\Gamma_A x}_u^2 = \normn{x}_A^2
\]

4
src/op/example/convolution.tex
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@@ -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}