Revisions.
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Bokuan Li
+13
-13
@@ -39,7 +39,7 @@
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\begin{lemma}
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\label{lemma:left-convolver-cc}
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Let $G$ be a locally compact group and $f \in C_c(G; \complex)$, $f$ is a lefy convolver.
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Let $G$ be a locally compact group and $f \in C_c(G; \complex)$, $f$ is a left convolver.
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\end{lemma}
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\begin{proof}
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By \hyperref[Young's Inequality]{theorem:young-fourier}, for each $g \in L^2(G; \complex)$, $\norm{f * g}_{L^2(G; \complex)} \le \norm{f}_{L^1(G; \complex)}\norm{g}_{L^2(G; \complex)}$.
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@@ -48,7 +48,7 @@
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\begin{lemma}
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\label{lemma:left-convolver}
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Let $G$ be a locally compacy group and $f: G \to \complex$ be a left convolver, then:
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Let $G$ be a locally compact group and $f: G \to \complex$ be a left convolver, then:
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\begin{enumerate}
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\item $L_f \in B(L^2(G; \complex))$.
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\item $f \in L_{\text{loc}}^1(G; \complex)$.
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@@ -64,8 +64,8 @@
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In which case, for almost every $x \in G$,
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\begin{align*}
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\limv{N} f * (g_N - g)(x) &= \int_G f(y)(g_N - g)(y^{-1}x)dy \\
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&\le \int_G|f(y)|\sum_{n \ge N}|(g_n - g)(y^{-1}x)| dy
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f * (g_N - g)(x) &= \int_G f(y)(g_N - g)(y^{-1}x)dy \\
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|f * (g_N - g)(x)| &\le \int_G|f(y)|\sum_{n \ge N}|(g_n - g)(y^{-1}x)| dy
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\end{align*}
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where $\int_G|f(y)|\sum_{n \in \natp}|(g_n - g)(y^{-1}x)| dy < \infty$ by (ii). As such, by the \hyperref[Dominated Convergence Theorem]{theorem:dct}, $f * g_N \to f * g$ almost everywhere. By (i), $f * g = h$ almost everywhere. Therefore the \hyperref[Closed Graph Theorem]{theorem:closed-graph} implies that $L_f$ is continuous.
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@@ -87,7 +87,7 @@
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\begin{definition}[Space of Left Convolvers]
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\label{definition:left-convolver-algebra}
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Let $G$ be a group, $LC(G) := \bracsn{L_f| G \to \complex\text{ is a left convolver}} \subset B(l^2(G; \complex))$, and $f, g: G \to \complex$ be left convolvers, then:
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Let $G$ be a group, $LC(G) := \bracsn{L_f| f: G \to \complex\text{ is a left convolver}} \subset B(l^2(G; \complex))$, and $f, g: G \to \complex$ be left convolvers, then:
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\begin{enumerate}
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\item $f \in l^2(G; \complex)$.
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\item $L_f^* = L_{f^*}$, where $f^*(x) = \ol{f(x^{-1})}$ for all $x \in G$.
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@@ -97,13 +97,13 @@
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The set $LC(G)$ is the \textbf{algebra of left convolvers} on $G$.
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\end{definition}
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\begin{proof}[Proof, {{\cite[Propositon 1.3.5]{AnantharamanPopa}}}. ]
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\begin{proof}[Proof, {{\cite[Proposition 1.3.5]{AnantharamanPopa}}}. ]
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(1): $f * \delta_e = f \in l^2(G; \complex)$.
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(2): Let $x, y \in G$, then
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\begin{align*}
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\dpn{L_f\delta_x, \delta_y}{l^2(G; \complex)} &= (f * \delta_x)(y) = f(x^{-1}y) \\
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\dpn{\delta_x, L_{f^*}\delta_y}{l^2(G;\complex)} &= \ol{(f^* * \delta_y)(x)} = f^*(y^{-1}x)
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\dpn{L_f\delta_x, \delta_y}{l^2(G; \complex)} &= (f * \delta_x)(y) = f(yx^{-1}) \\
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\dpn{\delta_x, L_{f^*}\delta_y}{l^2(G;\complex)} &= \ol{(f^* * \delta_y)(x)} = \ol{f^*(y^{-1}x)} = f(yx^{-1})
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\end{align*}
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@@ -114,7 +114,7 @@
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(4): Let $T \in \ol{LC(G)}^{\text{\small SOT}}$ and $\angles{f_\alpha}_{\alpha \in A}$ be left convolvers such that $T = \sotlim_{\alpha \in A}L_{f_\alpha}$. Let $f = T\delta_e \in l^2(G; \complex)$, then
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\[
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f = T\delta_e = \paren{\sotlim_{\alpha \in A}L_{f_\alpha}}\delta_e = l^2\text{-}\lim_{\alpha \in A}f_\alpha * e
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f = T\delta_e = \paren{\sotlim_{\alpha \in A}L_{f_\alpha}}\delta_e = l^2\text{-}\lim_{\alpha \in A}f_\alpha * \delta_e
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\]
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For any $g \in l^2(G; \complex)$,
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@@ -132,7 +132,7 @@
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\begin{proof}[Proof, {{\cite[Theorem 1.3.6]{AnantharamanPopa}}}. ]
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($\subset$): For any $x, y, z \in G$ and $f \in l^2(G; \complex)$,
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\[
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(\lambda(x)\rho(y)f)(z) = (\rho(h)f)(x^{-1}z) = f(x^{-1}zy) = (\lambda(x)f)(zy) = (\rho(y)\lambda(x)f)(z)
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(\lambda(x)\rho(y)f)(z) = (\rho(y)f)(x^{-1}z) = f(x^{-1}zy) = (\lambda(x)f)(zy) = (\rho(y)\lambda(x)f)(z)
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\]
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so $\lambda(G) \subset R(G)'$. By \autoref{definition:left-convolver-algebra}, $L(G) \subset LC(G) \subset R(G)'$.
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@@ -147,7 +147,7 @@
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\begin{definition}
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\label{definition:group-algebra-fourier-series}
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Let $G$ be a group. For each $x \in G$, let $u_x = \lambda(x)$ and identify it with $\delta_x$, then $\lambda(G) = \bracsn{u_x|x \in G}$ is the \textbf{canonical unitaries} of $L(G)$.
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Let $G$ be a group. For each $x \in G$, let $u_x = \lambda(x)$ and identify it with $\delta_x$, then $\lambda(G) = \bracsn{u_x|x \in G}$ are the \textbf{canonical unitaries} of $L(G)$.
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For each $f \in l^2(G)$, $f = \sum_{x \in G}f(x)u_x$. In particular, for each $T \in L(G) = LC(G) \subset l^2(G; \complex)$, there exists $\bracsn{T_x}_{x \in G} \subset \complex$ such that $T$ may be written as $T = \sum_{x \in G}T_x u_x$\footnote{This sum does not converge in the strong-operator or the weak-operator topology. }. In which case, the scalars $\bracsn{T_x}_{x \in G}$ are the \textbf{Fourier coefficients} of $T$.
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\end{definition}
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@@ -187,7 +187,7 @@
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\label{proposition:wreath-product-icc}
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Let $H$ be a non-trivial group, $\Gamma$ be a group acting on a set $I$, and $\Gamma_f \subset \Gamma$ be the subgroup of elements whose conjugacy class is finite. If
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\begin{enumerate}[label=(\alph*)]
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\item The orbits of $\Gamma \curvearrowleft I$ are finite.
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\item The orbits of $\Gamma \curvearrowleft I$ are infinite.
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\item The restricted action $\Gamma_f \curvearrowright I$ is faithful.
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\end{enumerate}
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@@ -198,7 +198,7 @@
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Let $g = (\xi, \gamma) \in H \wr_I \Gamma$. First suppose that $\xi \ne e_{H^{(I)}}$, then the support of $\xi$ is non-empty. By (a), the support of $\xi$ has infinite orbit under $\Gamma$, so $g^G \supset g^\Gamma$ is infinite.
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Now suppose that $\xi = e_{H^{(I)}}$ and $\gamma \ne e_{\Gamma}$. If $\gamma \in \Gamma \setminus \Gamma_f$, then the conjugacy class of $\gamma$ is infinite, and the conjugacy class of $g$ is also infinite. Now suppose that $\gamma \in \Gamma_f$. By (b), there exists $i \in I$ with $\gamma i_0 \ne i_0$. As $H$ is non-trivial, there exists $\xi_0 \in H^{(I)} \setminus \bracsn{e_{H^{(I)}}}$ with support $\bracsn{i_0}$. In which case, if $g_0 = (\xi_0, e_{\Gamma})$, then
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Now suppose that $\xi = e_{H^{(I)}}$ and $\gamma \ne e_{\Gamma}$. If $\gamma \in \Gamma \setminus \Gamma_f$, then the conjugacy class of $\gamma$ is infinite, and the conjugacy class of $g$ is also infinite. Now suppose that $\gamma \in \Gamma_f$. By (b), there exists $i_0 \in I$ with $\gamma i_0 \ne i_0$. As $H$ is non-trivial, there exists $\xi_0 \in H^{(I)} \setminus \bracsn{e_{H^{(I)}}}$ with support $\bracsn{i_0}$. In which case, if $g_0 = (\xi_0, e_{\Gamma})$, then
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\[
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g_0^{-1}gg_0 = (\xi_0^{-1}\gamma\xi_0, \gamma)
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\]
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