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Corrections round 1.

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Bokuan Li
2026-10-02 18:45:01 -04:00
parent a1a78d44d8
commit cd1499704f
1 changed files with 9 additions and 9 deletions
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src/harmonic/rep/pos.tex
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@@ -12,7 +12,7 @@
\begin{proof}
\begin{align*}
\int_G (f^* * f)(x)\phi(x) dx &= \iint_{G \times G} \Delta_G(y^{-1})\ol{f(y^{-1})}f(y^{-1}x)\phi(x)dydx \\
&= \iint_G \ol{f(y)} f(yx)\phi(x)dy dx \\
&= \iint_{G \times G} \ol{f(y)} f(yx)\phi(x)dy dx \\
&= \iint_{G \times G} f(x) \ol{f(y)} \phi(y^{-1}x)dydx
\end{align*}
\end{proof}
@@ -129,7 +129,7 @@
\begin{theorem}
\label{theorem:group-bc-weak-topology}
Let $G$ be a locally compact group, then the topology of uniform convergence on compact subsets of $G$ on $S(G)$ and the weak* topology on $S(G)$ as a subset of $L^1(G; \complex)^*$.
Let $G$ be a locally compact group, then the topology of uniform convergence on compact subsets of $G$ on $S(G)$ and the weak* topology on $S(G)$ as a subset of $L^1(G; \complex)^*$ coincide.
\end{theorem}
\begin{proof}
($\supset$): Let $\seqf{f_j} \subset L^1(G; \complex)$. By inner regularity of the Haar measure and hence $|f_j(x)|dx$, there exists $K \subset G$ compact such that $\int_{G \setminus K}|f_j| < \eps$ for all $1 \le j \le n$. In which case, for any $\phi, \psi \in \mathcal{F}$,
@@ -141,9 +141,9 @@
($\subset$): Let $\phi \in S(G)$ and $V \in \cn_G(1_G)$ be relatively compact, then for each $x \in G$,
\begin{align*}
\abs{\phi(x) - \frac{\one_V}{|V|} * \phi(x)} &\le \frac{1}{|V|}\int_V |\phi(x) - \phi(y^{-1}x)|dx \\
\abs{\phi(x) - \frac{\one_V}{|V|} * \phi(x)} &\le \frac{1}{|V|}\int_V |\phi(x) - \phi(y^{-1}x)|dy \\
&\le \frac{1}{|V|}\int_V ({2 - 2\text{Re}(\phi(y))})^{1/2} dy \\
&\le |V|^{1/2}\braks{\int_V 2 - 2\text{Re}(\phi(y)) dy}^{1/2}
&\le |V|^{-1/2}\braks{\int_V 2 - 2\text{Re}(\phi(y)) dy}^{1/2}
\end{align*}
by \autoref{lemma:lcg-positive-definite-function-concentration} and the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}. As such, for any $\phi, \psi \in S(G)$ and $x \in G$,
@@ -181,7 +181,7 @@
\label{theorem:lcg-state-extreme}
Let $G$ be a locally compact group, then:
\begin{enumerate}
\item The extreme points of $P_0(G)$ is the extreme points of $S(G)$ and $0$.
\item The extreme points of $P_0(G)$ are the extreme points of $S(G)$ and $0$.
\item $S(G)$ is contained in the closed convex hull of its extreme points\footnote{In general, $L^1(G; \complex)$ is not unital and $S(G)$ is not closed.}.
\end{enumerate}
\end{theorem}
@@ -191,13 +191,13 @@
1 = \norm{\phi}_{L^1(G; \complex)^*} = (1 - t)\norm{\psi_1}_{L^1(G; \complex)^*} + t\norm{\psi_2}_{L^1(G; \complex)^*} \le 1
\]
so $\norm{\psi_2}_{L^1(G; \complex)^*} = \norm{\psi_2}_{L^1(G; \complex)^*} = 1$, and $S(G)$ is an extreme subset of $P_0(G)$. By \autoref{lemma:extreme-subset-extreme-point}, every extreme point of $S(G)$ is an extreme point of $P_0(G)$.
so $\norm{\psi_1}_{L^1(G; \complex)^*} = \norm{\psi_2}_{L^1(G; \complex)^*} = 1$, and $S(G)$ is an extreme subset of $P_0(G)$. By \autoref{lemma:extreme-subset-extreme-point}, every extreme point of $S(G)$ is an extreme point of $P_0(G)$.
On the other hand, every extreme point of $P_0(G)$ in $S(G)$ is an extreme point of $S(G)$.
It remains to show that $0$ is the only other extreme point of $P_0(G)$. To this end, let $\phi, \psi \in P_0(G)$ and $t \in (0, 1)$ such that $0 = (1 - t)\phi + t\psi$, then $0 = (1 - t)\phi(1_G) + t\psi(1_G)$. By \autoref{corollary:group-l1-positive-function}, $0 = \phi(1_G) = \norm{\phi}_u$ and $0 = \psi(1_G) = \norm{\psi}_u$ so $0$ is an extreme point of $P_0(G)$. For any $\phi \in P_0(G) \setminus (S(G) \cup \bracsn{0})$, $\phi/\phi(1_G) \in S(G)$, and $\phi = 0 + \phi(1_G) \cdot \phi/\phi(1_G)$. Therefore $0$ is the only other extreme point of $P_0(G)$.
(2): Let $\phi \in S(G)$ and $\eps \in (0, 1)$. By definition of the operator norm and (4) of \autoref{proposition:semicontinuous-properties}, $\norm{\cdot}_{L^1(G; \complex)^*}$ is lower semicontinuous on $L^1(G; \complex)^*$ with respect to the weak topology, so $\bracsn{\psi \in P_0(G)|\ \norm{\psi}_{L^1(G; \complex)^*} > 1 - \eps}$ is a weak*-neighbourhood of $\phi$.
(2): Let $\phi \in S(G)$ and $\eps \in (0, 1)$. By definition of the operator norm and (4) of \autoref{proposition:semicontinuous-properties}, $\norm{\cdot}_{L^1(G; \complex)^*}$ is lower semicontinuous on $L^1(G; \complex)^*$ with respect to the weak* topology, so $\bracsn{\psi \in P_0(G)|\ \norm{\psi}_{L^1(G; \complex)^*} > 1 - \eps}$ is a weak*-neighbourhood of $\phi$.
Let $\cf \subset L^1(G; \complex)$ be finite, then by (1) and the \hyperref[Krein-Milman Theorem]{theorem:krein-milman}, there exists $\seqf{\phi_j} \subset S(G)$ and $\bracsn{\lambda_j}_0^n \subset [0, 1]$ such that $\sum_{j = 0}^n \lambda_j = 1$, $\lambda_0 < \eps/\max_{f \in \cf}\norm{f}_{L^1(G; \complex)}$, and
\[
@@ -221,13 +221,13 @@
Let $G$ be a locally compact group and $x, y \in G$ be distinct, then there exists an irreducible unitary representation $(H, \pi)$ of $G$ such that $\pi(x) \ne \pi(y)$.
\end{theorem}
\begin{proof}[Proof, {{\cite[Theorem 3.34]{FollandHarmonic}}}. ]
Let $x, y \in G$ and suppose that for any irreducible unitary representation of $(H, \pi)$ of $G$, $\pi(x) = \pi(y)$. Let $\phi \in S(G)$ be an extreme point of $S(G)$ and $(H_\phi, \pi_\phi, \xi_\phi)$ be its \hyperref[GNS triple]{definition:gns-triple-lcg}, then $(H_\phi, \pi_\phi)$ is irreducible by \autoref{proposition:gns-irreducible-lcg}. In particular, by \autoref{corollary:group-l1-positive-function},
Let $x, y \in G$ and suppose that for any irreducible unitary representation $(H, \pi)$ of $G$, $\pi(x) = \pi(y)$. Let $\phi \in S(G)$ be an extreme point of $S(G)$ and $(H_\phi, \pi_\phi, \xi_\phi)$ be its \hyperref[GNS triple]{definition:gns-triple-lcg}, then $(H_\phi, \pi_\phi)$ is irreducible by \autoref{proposition:gns-irreducible-lcg}. In particular, by \autoref{corollary:group-l1-positive-function},
\[
\phi(x) = \dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi} =
\dpn{\pi_\phi(y)\xi_\phi, \xi_\phi}{H_\phi} = \phi(y)
\]
By (2) of \autoref{theorem:lcg-state-extreme}, the extreme points of $S(G)$ is weak*-dense in $S(G)$. By \autoref{theorem:group-bc-weak-topology}, the weak* topology and the topology of uniform convergence on compact subsets of $G$ coincide on $S(G)$. As such, $\psi(x) = \psi(y)$ for all $\psi \in S(G)$. By linearity, $\psi(x) = \psi(y)$ for all $\psi \in P(G)$ as well.
By (2) of \autoref{theorem:lcg-state-extreme}, the extreme points of $S(G)$ are weak*-dense in $S(G)$. By \autoref{theorem:group-bc-weak-topology}, the weak* topology and the topology of uniform convergence on compact subsets of $G$ coincide on $S(G)$. As such, $\psi(x) = \psi(y)$ for all $\psi \in S(G)$. By linearity, $\psi(x) = \psi(y)$ for all $\psi \in P(G)$ as well.
By linearity and \autoref{lemma:positive-definite-cc-decompose}, $(f * g)(x) = (f * g)(y)$ for all $f, g \in C_c(G; \complex)$. Using \autoref{proposition:group-algebra-approximate-identity}, let $\angles{\psi_U}_{U \in \cn_G(1_G)} \subset C_c(G; \complex)$ be an approximate identity for $L^1(G; \complex)$ such that $\supp{\psi_U} \subset U$ for all $U \in \cn_G(1_G)$, then for any $g \in C_c(G; \complex)$, $\psi_U * g \to g$ uniformly. Therefore $g(x) = g(y)$ for all $g \in C_c(G; \complex)$.