From 2d09d41e7513d1549df2cceb997e565719f0139a Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sat, 26 Sep 2026 15:05:00 -0400 Subject: [PATCH] Revision. --- src/harmonic/rep/l1.tex | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/src/harmonic/rep/l1.tex b/src/harmonic/rep/l1.tex index 062dab8..fd5f5fa 100644 --- a/src/harmonic/rep/l1.tex +++ b/src/harmonic/rep/l1.tex @@ -11,14 +11,14 @@ \pi: L^1(G; \complex) \to B(H) \quad \dpn{\pi(f)\xi, \eta}{H} = \int_G f(x)\dpn{\pi_0(x)\xi, \eta}{H}dx \] - defines a non-degenerate *-representation of $G$. + defines a non-degenerate *-representation of $L^1(G; \complex)$. \item For any $x \in G$ and $f \in L^1(G; \complex)$, \[ - \pi(x)\pi(f) = \pi(L_xf) \quad \pi(f)\pi(x) = \Delta_G(x^{-1})\pi(R_{x^{-1}}f) + \pi_0(x)\pi(f) = \pi(L_xf) \quad \pi(f)\pi_0(x) = \Delta_G(x^{-1})\pi(R_{x^{-1}}f) \] \end{enumerate} - Conversely, if $\pi: L^1(G; \complex) \to B(H)$ is a non-degenerate *-representation of $G$, then: + Conversely, if $\pi: L^1(G; \complex) \to B(H)$ is a non-degenerate *-representation of $L^1(G; \complex)$, then: \begin{enumerate}[start=2] \item There exists a unique unitary representation $\pi_0: G \to B(H)$ such that \[ @@ -40,7 +40,7 @@ (1, Homomorphism): For any $f, g \in L^1(G; \complex)$ and $\xi, \eta \in H$, since $\pi_0$ is a unitary representation, \begin{align*} - \dpn{\pi(f * g)\xi, \eta}H &= \int_G (f * g)(x)\dpn{\pi_0(x)\xi, \eta}{H}dx \\ + \dpn{\pi(f * g)\xi, \eta}{H} &= \int_G (f * g)(x)\dpn{\pi_0(x)\xi, \eta}{H}dx \\ &= \iint_{G \times G}f(y)g(y^{-1}x)\dpn{\pi_0(x)\xi, \eta}{H}dydx \\ &= \int_{G}f(y)\int_Gg(x)\dpn{\pi_0(y)\pi_0(x)\xi, \eta}{H}dxdy\\ &= \int_{G}f(y)\int_Gg(x)\dpn{\pi_0(x)\xi, \pi_0(y^{-1})\eta}{H}dxdy \\ @@ -62,21 +62,21 @@ |\dpn{\pi(f)\xi - \xi, \eta}{H}| &\le \frac{1}{|V|}\int_V \norm{\pi(x)\xi - \xi}_H\norm{\eta}_Hdx < \norm{\xi}_H\norm{\eta}_H \end{align*} - so $\norm{\pi(f)\xi - \xi}_H < \norm{\xi}$, and $\pi(f)\xi \ne 0$. + so $\norm{\pi(f)\xi - \xi}_H < \norm{\xi}_H$, and $\pi(f)\xi \ne 0$. (3): Using \autoref{proposition:group-algebra-approximate-identity}, let $\angles{\phi_\alpha}_{\alpha \in A}$ be an approximate identity in $L^1(G; \complex)$. For each $x \in G$ and $f \in L^1(G; \complex)$, $L_x\phi_\alpha * f \to L_x f$ in $L^1(G; \complex)$ implies that $\angles{\pi(\phi_\alpha)}_{\alpha \in A}$ converges pointwise on \[ K := \text{span}\bracsn{\pi(g)\xi|\xi \in H, g \in L^1(G; \complex)} \] - Given that $\pi$ is non-degenerate, the $K$ is dense in $H$. As $\bracsn{L_x\phi_\alpha|\alpha \in A}$ is equicontinuous and $H$ is complete, the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli} implies that there exists $\pi_0(x) \in \ol{B_{B(H)}(0, 1)}$ such that $L_x\phi_\alpha \to \pi_0(x)$ in the strong operator topology. In particular, $\pi_0(1) = \text{Id}$. + Given that $\pi$ is non-degenerate, $K$ is dense in $H$. As $\bracsn{L_x\phi_\alpha|\alpha \in A}$ is equicontinuous and $H$ is complete, the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli} implies that there exists $\pi_0(x) \in \ol{B_{B(H)}(0, 1)}$ such that $L_x\phi_\alpha \to \pi_0(x)$ in the strong operator topology. In particular, $\pi_0(1) = \text{Id}$. For each $f \in L^1(G; \complex)$, $\xi \in H$, and $x, y \in G$, \[ \pi_0(xy)\pi(f)\xi = \pi(L_{xy}f)\xi = \pi(L_xL_yf)\xi = \pi_0(x)\pi_0(y)\pi(f)\xi \] - By density of $K$ in $H$, $\pi_0(x)\pi_0(y) = \pi_0(x)\pi_0(y)$, so $\pi$ is a homomorphism. For any $x \in G$, $\text{Id} = \pi(1) = \pi(x)\pi(x^{-1})$, so $\pi(x)$ is isometric and invertible, and hence unitary. + By density of $K$ in $H$, $\pi_0(xy) = \pi_0(x)\pi_0(y)$, so $\pi$ is a homomorphism. For any $x \in G$, $\text{Id} = \pi_0(1) = \pi_0(x)\pi_0(x^{-1})$, so $\pi_0(x)$ is isometric and invertible, and hence unitary. Finally, for any $f \in L^1(G; \complex)$ and $\xi, \eta \in H$, by the \hyperref[Dominated Convergence Theorem]{theorem:dct}, \begin{align*} @@ -93,11 +93,11 @@ \begin{proof} By (3) of \autoref{definition:representation-l1-group-algebra}, $\pi_0(G)$ lies in the strong-operator closure of $L^1(G; \complex)$. - On the other hand, let $K \subset G$ be compact, $\eps > 0$, and $\seqf{(\xi_j, \eta_j)} \subset H^2$, then there exists non-empty sets $\bracsn{V_k}_1^m \subset 2^G$ such that $K = \bigsqcup_{j = 1}^n V_j$ and $|\dpn{[\pi_0(x) - \pi_0(y)]\xi_j, \eta_j}{H}| \le \eps$ for all $x, y \in V_k$, $1 \le j \le n$, and $1 \le k \le m$. + On the other hand, let $K \subset G$ be compact, $\eps > 0$, and $\seqf{(\xi_j, \eta_j)} \subset H^2$, then there exist non-empty sets $\bracsn{V_k}_1^m \subset 2^G$ such that $K = \bigsqcup_{j = 1}^n V_j$ and $|\dpn{[\pi_0(x) - \pi_0(y)]\xi_j, \eta_j}{H}| \le \eps$ for all $x, y \in V_k$, $1 \le j \le n$, and $1 \le k \le m$. Let $\seqf{x_k} \subset G$ with $x_k \in V_k$ for all $1 \le k \le m$, then \begin{align*} - \abs{\angles{\braks{\pi(\one_K) - \sum_{k = 1}^n |V_k|\pi_0(x_k)}\xi_j, \eta_j}_{H}} &\le \sum_{k = 1}^m \int_{V_k}|\dpn{\pi_0(y) - \pi_0(x_k)\xi_j, \eta_j}{H}|dy \\ + \abs{\angles{\braks{\pi(\one_K) - \sum_{k = 1}^m |V_k|\pi_0(x_k)}\xi_j, \eta_j}_{H}} &\le \sum_{k = 1}^m \int_{V_k}|\dpn{[\pi_0(y) - \pi_0(x_k)]\xi_j, \eta_j}{H}|dy \\ &\le \sum_{k = 1}^m \eps |V_k| = \eps |K| \end{align*}