Oopsies forgot a theorem.

src/harmonic/lcg/algebra.tex

@@ -306,3 +306,28 @@ Unfortunately, $L^1(G)$ is not a $C^*$-algebra, and its approximate identity req
     and $f * \phi_U \to f$ in $L^p(G; \complex)$ by \hyperref[continuity of translation]{proposition:haar-translation}.
 \end{proof}
 
+\begin{theorem}
+\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 $x \in G$. 
+\end{theorem}
+\begin{proof}
+    ($\Rightarrow$): 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 $x \in G$,
+    \[
+    L_xf = \lim_{\alpha \in A}L_x(\phi_\alpha * f) = \lim_{\alpha \in A}(L_x\phi_\alpha) * f \in J
+    \]
+
+    ($\Leftarrow$): 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*}
+        (\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) - \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 \\
+        \abs{(\one_A * f) - \sum_{y \in F}|A \cap U_j|L_yf} &\le \eps |A|
+    \end{align*}
+
+    where for any $B \in \cb_G$, $|B|$ denotes its measure. As the above holds for all $\eps > 0$, $\one_A * f \in \ol J = J$. 
+
+    For any Borel set $B \in \cb_G$ with finite measure and $\eps > 0$, by inner regularity of the Haar measure, there exists $A \subset G$ compact such that $\norm{\one_B - \one_A}_{L^1(G; \complex)} < \eps$. By continuity of the convolution product, $\one_B * f \in J$ as well. 
+
+    By linearity, $\phi * f \in J$ for any $\phi \in \Sigma(G; \complex) \cap L^1(G; \complex)$. Finally, by \hyperref[density of simple functions]{proposition:lp-simple-dense}, $g * f \in J$ for all $g \in L^1(G; \complex)$. Therefore $J$ is a left ideal. 
+\end{proof}
+
+