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Typo fixes.

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Bokuan Li
2026-09-23 16:00:47 -04:00
parent 0a17a5a4a1
commit 03e0118d53
1 changed files with 4 additions and 4 deletions
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src/harmonic/lcg/algebra.tex
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@@ -310,7 +310,7 @@ Unfortunately, $L^1(G)$ is not a $C^*$-algebra, and its approximate identity req
\label{theorem:lcg-group-algebra-ideal} \label{theorem:lcg-group-algebra-ideal}
Let $G$ be a locally compact group and $J \subset L^1(G; \complex)$ be a closed subspace, then $J$ is a left ideal if and only if $L_xJ \subset J$ for all $x \in G$. Let $G$ be a locally compact group and $J \subset L^1(G; \complex)$ be a closed subspace, then $J$ is a left ideal if and only if $L_xJ \subset J$ for all $x \in G$.
\end{theorem} \end{theorem}
\begin{proof} \begin{proof}[Proof, {{\cite[Theorem 2.45]{FollandHarmonic}}}. ]
($\Rightarrow$): Suppose that $J$ is a left ideal. By \autoref{proposition:group-algebra-approximate-identity}, $L^1(G; \complex)$ admits an approximate identity $\angles{\phi_\alpha}_{\alpha \in A}$. In which case, for any $f \in J$ and $x \in G$, ($\Rightarrow$): Suppose that $J$ is a left ideal. By \autoref{proposition:group-algebra-approximate-identity}, $L^1(G; \complex)$ admits an approximate identity $\angles{\phi_\alpha}_{\alpha \in A}$. In which case, for any $f \in J$ and $x \in G$,
\[ \[
L_xf = \lim_{\alpha \in A}L_x(\phi_\alpha * f) = \lim_{\alpha \in A}(L_x\phi_\alpha) * f \in J L_xf = \lim_{\alpha \in A}L_x(\phi_\alpha * f) = \lim_{\alpha \in A}(L_x\phi_\alpha) * f \in J
@@ -318,9 +318,9 @@ Unfortunately, $L^1(G)$ is not a $C^*$-algebra, and its approximate identity req
($\Leftarrow$): Suppose that $L_xJ \subset J$ for all $x \in G$. Let $f \in J$, $A \subset G$ be compact, and $\eps > 0$. By \autoref{proposition:haar-translation}, for each $y \in A$, there exists $U_y \in \cn_G(y)$ such that $\norm{L_yf - L_zf}_{L^1(G; \complex)} \le \eps$ for all $z \in U_y$. By compactness of $A$, there exists $F \subset A$ finite such that $A \subset \bigcup_{y \in F}U_y$. In which case, ($\Leftarrow$): Suppose that $L_xJ \subset J$ for all $x \in G$. Let $f \in J$, $A \subset G$ be compact, and $\eps > 0$. By \autoref{proposition:haar-translation}, for each $y \in A$, there exists $U_y \in \cn_G(y)$ such that $\norm{L_yf - L_zf}_{L^1(G; \complex)} \le \eps$ for all $z \in U_y$. By compactness of $A$, there exists $F \subset A$ finite such that $A \subset \bigcup_{y \in F}U_y$. In which case,
\begin{align*} \begin{align*}
(\one_A * f)(x) &= \int_A L_yf(x)dy = \sum_{y \in F}\int_{A \cap U_j} L_zf(x)dz \\ (\one_A * f)(x) &= \int_A L_yf(x)dy = \sum_{y \in F}\int_{A \cap U_y} L_zf(x)dz \\
(\one_A * f)(x) - \sum_{y \in F}|A \cap U_j|L_yf(x) &= \sum_{y \in F}\int_{A \cap U_j} (L_zf - L_yf)(x)dz \\ (\one_A * f)(x) - \sum_{y \in F}|A \cap U_y|L_yf(x) &= \sum_{y \in F}\int_{A \cap U_y} (L_zf - L_yf)(x)dz \\
\abs{(\one_A * f) - \sum_{y \in F}|A \cap U_j|L_yf} &\le \eps |A| \abs{(\one_A * f) - \sum_{y \in F}|A \cap U_y|L_yf} &\le \eps |A|
\end{align*} \end{align*}
where for any $B \in \cb_G$, $|B|$ denotes its measure. As the above holds for all $\eps > 0$, $\one_A * f \in \ol J = J$. where for any $B \in \cb_G$, $|B|$ denotes its measure. As the above holds for all $\eps > 0$, $\one_A * f \in \ol J = J$.