diff --git a/src/harmonic/index.tex b/src/harmonic/index.tex index 714fa52..bb76797 100644 --- a/src/harmonic/index.tex +++ b/src/harmonic/index.tex @@ -2,4 +2,5 @@ \label{part:harmonic} \input{./lcg/index.tex} +\input{./rep/index.tex} \input{./notation.tex} diff --git a/src/harmonic/rep/index.tex b/src/harmonic/rep/index.tex new file mode 100644 index 0000000..a31315e --- /dev/null +++ b/src/harmonic/rep/index.tex @@ -0,0 +1,6 @@ +\chapter{Representation of Locally Compact Groups} +\label{chap:lcg-rep} + +\input{./unit.tex} +\input{./l1.tex} + diff --git a/src/harmonic/rep/l1.tex b/src/harmonic/rep/l1.tex new file mode 100644 index 0000000..062dab8 --- /dev/null +++ b/src/harmonic/rep/l1.tex @@ -0,0 +1,108 @@ +\section{Representations of $L^1(G)$} +\label{section:rep-l1-group-algebra} + + +\begin{definition}[Representation of $L^1(G; \complex)$] +\label{definition:representation-l1-group-algebra} + Let $G$ be a locally compact group and $(H, \pi_0)$ be a unitary representation of $G$, then + \begin{enumerate} + \item The mapping + \[ + \pi: L^1(G; \complex) \to B(H) \quad \dpn{\pi(f)\xi, \eta}{H} = \int_G f(x)\dpn{\pi_0(x)\xi, \eta}{H}dx + \] + + defines a non-degenerate *-representation of $G$. + \item For any $x \in G$ and $f \in L^1(G; \complex)$, + \[ + \pi(x)\pi(f) = \pi(L_xf) \quad \pi(f)\pi(x) = \Delta_G(x^{-1})\pi(R_{x^{-1}}f) + \] + \end{enumerate} + + Conversely, if $\pi: L^1(G; \complex) \to B(H)$ is a non-degenerate *-representation of $G$, then: + \begin{enumerate}[start=2] + \item There exists a unique unitary representation $\pi_0: G \to B(H)$ such that + \[ + \dpn{\pi(f)\xi, \eta}{H} = \int_G f(x)\dpn{\pi_0(x)\xi, \eta}{H}dx + \] + + for all $f \in L^1(G; \complex)$ and $\xi, \eta \in H$. Moreover, for any approximate identity $\angles{\phi_\alpha}_{\alpha \in A} \subset L^1(G; \complex)$ and $x \in G$, $\pi_0(x) = \sotlim_{\alpha \in A}\pi(L_x\phi_\alpha)$. + \end{enumerate} + + The mapping $\pi: L^1(G; \complex) \to B(H)$ is the \textbf{representation of $L^1(G; \complex)$ determined by $\pi_0$}. +\end{definition} +\begin{proof}[Proof, {{\cite[Theorem 3.9, Theorem 3.11]{FollandHarmonic}}}. ] + (1, Well-Defined): For each $x \in G$, $\pi_0(x)$ is unitary, so for any $f \in L^1(G; \complex)$ and $\xi, \eta \in H$, + \[ + \abs{\dpn{\pi(f)\xi, \eta}{H}} \le \int_G \abs{f(x)\dpn{\pi_0(x)\xi, \eta}{H}}dx \le \norm{\xi}_H\norm{\eta}_H\norm{f}_{L^1(G; \complex)} + \] + + and the representation is well-defined. + + (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 \\ + &= \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 \\ + &= \int_G f(y) \dpn{\pi(g)\xi, \pi_0(y^{-1})\eta}{H} dy\\ + &= \int_G f(y) \dpn{\pi_0(y)\pi(g)\xi,\eta}{H} dy = \dpn{\pi(f)\pi(g)\xi, \eta}{H} + \end{align*} + + (1, *-Homomorphism): For any $f \in L^1(G; \complex)$ and $\xi, \eta \in H$, since $\pi_0$ is unitary, + \begin{align*} + \dpn{\pi(f^*)\xi, \eta}{H} &= \int_G \Delta_G(x^{-1})\ol{f(x^{-1})}\dpn{\pi_0(x)\xi, \eta}{H} dx \\ + &= \int_G \ol{f(x)} \dpn{\pi_0(x^{-1})\xi, \eta}{H}dx = \int_G \ol{f(x)} \dpn{\xi, \pi_0(x)\eta}{H}dx \\ + &= \int_G \ol{f(x)} \cdot \ol{\dpn{\pi_0(x)\eta, \xi}{H}} dx = \overline{\int_G f(x)\dpn{\pi_0(x)\eta, \xi}{H} dx} \\ + &= \ol{\dpn{\pi(f)\eta, \xi}{H}} = \dpn{\xi, \pi(f)\eta}{H} + \end{align*} + + (1, Non-Degenerate): For any $\xi \in H \setminus \bracsn{0}$, since $\pi_0$ is strong-operator continuous, there exists $V \in \cn_G(1)$ such that $\norm{\pi(x)\xi - \xi}_H < \norm{\xi}_H$ for all $x \in V$. In which case, if $f = |V|^{-1}\one_V$\footnote{$|V|$ is the value of the Haar measure at $V$.}, then for any $\eta \in H$, + \begin{align*} + \dpn{\pi(f)\xi - \xi, \eta}{H} &= \frac{1}{|V|}\int_V \dpn{\pi(x)\xi - \xi, \eta}{H} dx \\ + |\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 $\pi(f)\xi \ne 0$. + + (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, $\pi_0(1) = \text{Id}$. + + 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. + + Finally, for any $f \in L^1(G; \complex)$ and $\xi, \eta \in H$, by the \hyperref[Dominated Convergence Theorem]{theorem:dct}, + \begin{align*} + \dpn{\pi(f)\pi(\phi_\alpha)\xi, \eta}{H} &= \dpn{\pi(f * \phi_\alpha)\xi, \eta}{H} = \angles{\pi\paren{\int_G f(y)L_y\phi_\alpha dy}\xi, \eta}_{H} \\ + &= \int_G f(y) \dpn{\pi(L_y\phi_\alpha)\xi, \eta}{H}dy \\ + \dpn{\pi(f)\xi, \eta}{H} &= \int_G f(y)\dpn{\pi_0(y)\xi, \eta}{H}dy + \end{align*} +\end{proof} + +\begin{theorem} +\label{theorem:l1-group-von-neumann} + Let $G$ be a locally compact group, $(H, \pi_0)$ be a unitary representation of $G$, and $\pi: L^1(G; \complex) \to B(H)$ be the *-representation determined by $\pi_0$, then the von Neumann algebras generated by $\pi_0(G)$ and $\pi(L^1(G; \complex))$ coincide. +\end{theorem} +\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 $1 \le k \le m$. + + 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 \\ + &\le \sum_{k = 1}^m \eps |V_k| = \eps |K| + \end{align*} + + so $\pi(\one_K)$ is in the von Neumann algebra generated by $\pi_0(G)$. By inner regularity of the Haar measure, $\pi(\one_A)$ is in the von Neumann algebra generated by $\pi_0(G)$ for all $A \in \cb_G$ with $\mu(A) < \infty$. By \hyperref[density of simple functions]{proposition:lp-simple-dense}, $\pi(L^1(G; \complex))$ is in the von Neumann algebra generated by $\pi_0(G)$. + +\end{proof} + + diff --git a/src/harmonic/rep/unit.tex b/src/harmonic/rep/unit.tex new file mode 100644 index 0000000..6c5060e --- /dev/null +++ b/src/harmonic/rep/unit.tex @@ -0,0 +1,122 @@ +\section{Unitary Representations of Locally Compact Groups} +\label{section:lcg-unitary} + +\begin{definition}[Unitary Representation] +\label{definition:lcg-unitary} + Let $G$ be a locally compact group, then a \textbf{unitary representation} of $G$ is a pair $(H, \pi)$, where $H$ is a non-zero complex Hilbert space and $\pi: G \to U(B(H))$ is a strong-operator continuous homomorphism. +\end{definition} + +\begin{definition}[Left Regular Representation] +\label{definition:lcg-left-regular} + Let $G$ be a locally compact group, then the mapping + \[ + \pi_L: G \to U(B(L^2(G))) \quad x \mapsto L_x + \] + + is the \textbf{left regular representation} of $G$. +\end{definition} + +\begin{definition}[Intertwining Operator] +\label{definition:lcg-intertwining} + Let $G$ be a locally compact group, $(H_1, \pi_1)$ and $(H_2, \pi_2)$ be unitary representations of $G$, and $T \in L(H_1; H_2)$, then $T$ is an \textbf{intertwining operator} for $\pi_1$ and $\pi_2$ if the following diagram commutes + \[ + \xymatrix{ + H_1 \ar@{->}[d]_{\pi_1(x)} \ar@{->}[r]^{T} & H_2 \ar@{->}[d]^{\pi_1(x)} \\ + H_1 \ar@{->}[r]_{T} & H_2 + } + \] + + for all $x \in G$. The set $\mathcal{C}(\pi_1, \pi_2)$ is the \textbf{space of intertwining operators} for $\pi_1$ and $\pi_2$. +\end{definition} + +\begin{lemma} +\label{lemma:lcg-intertwining} + Let $G$ be a locally compact group and $(H, \pi)$ be a unitary representation of $G$, then $\mathcal{C}(\pi, \pi)$ is the commutant of $\pi(G)'$, and hence a von Neumann algebra. +\end{lemma} + +\begin{definition}[Invariant Subspace] +\label{definition:lcg-invariant-subspace} + Let $G$ be a locally compact group, $(H, \pi)$ be a unitary representation of $G$, $M \subset H$ be a closed subspace, and $P \in B(H)$ be the orthogonal projection onto $M$, then the following are equivalent: + \begin{enumerate} + \item For each $x \in G$, $\pi(x)(M) \subset M$. + \item $P \in \pi(G)'$. + \end{enumerate} + + If the above holds, then $M$ is a \textbf{invariant subspace} for $\pi$. +\end{definition} +\begin{proof} + Since $\pi$ is unitary, $\pi(x)^* = \pi(x^{-1})$ for all $x \in G$. As such, $\pi(G)$ is a self-adjoint subset of $G$, so $M$ is invariant for $\pi(G)$ if and only if it is a reducing subspace for $\pi(G)$, if and only if $\pi(G)$ commutes with $P$. +\end{proof} + +\begin{definition}[Subrepresentation] +\label{definition:lcg-subrepresentation} + Let $G$ be a locally compact group, $(H, \pi)$ be a unitary representation of $G$, and $M \subset H$ be an invariant subspace, then the mapping + \[ + \pi_M: G \to U(B(M)) \quad x \mapsto \pi(x)|_M + \] + + is a \textbf{unitary subrepresentation} of $G$. + + If $\pi$ admits a non-trivial invariant subspace, then $\pi$ is \textbf{reducible}. Otherwise, it is \textbf{irreducible}. +\end{definition} + +\begin{definition}[Direct Sum] +\label{definition:lcg-rep-direct-sum} + Let $G$ be a locally compact group, $\bracsn{(H_i, \pi_i)}_{i \in I}$ be unitary representations of $G$, then the unitary representation + \[ + \pi: G \to B([l^2(I); H_i]) \quad \pi(x)(\xi)_i = \pi_i(x)(\xi_i) + \] + + is the \textbf{direct sum} of $\bracsn{(H_i, \pi_i)}_{i \in I}$, denoted $\bigoplus_{i \in I}\pi_i$. +\end{definition} + +\begin{definition}[Cyclic] +\label{definition:lcg-rep-cyclic} + Let $G$ be a locally compact group, $(H, \pi)$ be a unitary representation of $G$, and $\xi \in H$, then $\xi$ is a \textbf{cyclic vector} if $\text{span}\bracsn{\pi(x)\xi|x \in G}$ is dense in $H$. The representation $(H, \pi)$ is \textbf{cyclic} if it admits a cyclic vector. +\end{definition} + + + +\begin{proposition} +\label{proposition:lcg-rep-invariant-gymnastics} + Let $G$ be a locally compact group, $(H, \pi)$ be a unitary representation of $G$, and $M \subset H$ be an invariant subspace, then: + \begin{enumerate} + \item $M^\perp$ is also an invariant subspace. + \item $(H, \pi)$ is the direct sum of $(M, \pi_M)$ and $(M^\perp, \pi_{M^\perp})$. + \end{enumerate} + + Moreover, + \begin{enumerate}[start=3] + \item $(H, \pi)$ is a direct sum of cyclic representations. + \end{enumerate} +\end{proposition} +% Proof omitted. + +\begin{lemma}[Schur] +\label{lemma:lcg-schur} + Let $G$ be a locally compact group and $(H, \pi), (K, \tau)$ be unitary representations of $G$, then + \begin{enumerate} + \item $(H, \pi)$ is irreducible if and only if the commutant $\pi(G)' = \complex I$. + \item If $(H, \pi)$ and $(K, \tau)$ are irreducible, then $(H, \pi)$ and $(K, \tau)$ are unitarily equivalent if and only if $\mathcal{C}(\pi, \tau)$ is one-dimensional. Otherwise, $\mathcal{C}(\pi, \tau) = \bracsn{0}$. + \end{enumerate} +\end{lemma} +\begin{proof}[Proof, {{\cite[Theorem 3.5]{FollandHarmonic}}}. ] + (1): $(H, \pi)$ is irreducible if and only if $\pi(G)'$ contains no non-trivial projections. Since $\pi(G)'$ is a von Neumann algebra, \autoref{theorem:vn-projection-norm-dense} implies that the span of projections in $\pi(G)'$ is dense in $\pi(G)'$. Therefore $(H, \pi)$ is irreducible if and only if $\pi(G)' = \complex I$. + + (2): For any $T \in \mathcal{C}(\pi, \tau)$, $T^* \in \mathcal{C}(\tau, \pi)$, so $T^*T \in \mathcal{C}(\pi, \pi) = \pi(G)'$, and $TT^* \in \mathcal{C}(\tau, \tau) = \tau(G)'$. Given that $(H, \pi)$ and $(K, \tau)$ are irreducible, there exists $\lambda, \mu \in \complex$ such that $T^*T = \lambda I_H$ and $TT^* = \mu I_K$. As such, either $T = 0$ or $\lambda^{-1/2}T$\footnote{Pick any branch.} is unitary, so $\mathcal{C}(\pi, \tau)$ is non-zero if and only if $(H, \pi)$ and $(K, \tau)$ are unitarily equivalent. + + If $\mathcal{C}(\pi, \tau) \ne \bracsn{0}$, then $\mathcal{C}(\pi, \tau)$ consists of scalar multiples of unitary operators. In which case, for any unitary operators $S, T \in \mathcal{C}(\pi, \tau)$, $S^{-1}T = S^*T \in \mathcal{C}(\pi)$. Thus there exists $\lambda \in \complex$ such that $S^{-1}T = \lambda I_H$, so $T = \lambda S$, and $\mathcal{C}(\pi, \tau)$ is one-dimensional. +\end{proof} + +\begin{corollary} +\label{corollary:lcg-schur-abelian-irrep} + Let $G$ be a locally compact group. If $G$ is abelian, then every irreducible representation of $G$ is one-dimensional. +\end{corollary} +\begin{proof}[Proof, {{\cite[Corollary 3.6]{FollandHarmonic}}}. ] + Let $(H, \pi)$ be an irreducible representation of $G$. Since $G$ is abelian, $\pi(G) \subset \pi(G)'$. By \hyperref[Schur's Lemma]{lemma:lcg-schur}, $\pi(G)' = \complex I$. As every one-dimensional subspace of $H$ is invariant, $H$ itself must be one-dimensional. +\end{proof} + + + + + diff --git a/src/op/c-star/homomorphism.tex b/src/op/c-star/homomorphism.tex index 0e3b315..1fa99c7 100644 --- a/src/op/c-star/homomorphism.tex +++ b/src/op/c-star/homomorphism.tex @@ -17,6 +17,40 @@ \end{definition} + +\begin{definition}[*-Representation] +\label{definition:star-rep} + Let $A$ be an involutive Banach algebra, then a \textbf{*-representation} of $A$ is a pair $(H, \pi)$, where $H$ is a complex Hilbert space, and $\pi: A \to B(H)$ is a *-homomorphism. +\end{definition} + + +\begin{definition}[Non-Degenerate] +\label{definition:star-rep-nondegen} + Let $A$ be an involutive Banach algebra and $(H, \pi)$ be a *-representation of $A$, then the following are equivalent: + \begin{enumerate} + \item $\text{span}\bracsn{\pi(x)\xi|x \in A, \xi \in H}$ is dense in $H$. + \item There exists no non-zero $\xi \in H$ such that $\pi(x)\xi = 0$ for all $x \in A$. + \end{enumerate} + + If the above holds, then $(H, \pi)$ is \textbf{non-degenerate}. +\end{definition} +\begin{proof} + $\neg (1) \Rightarrow \neg (2)$: Let $\xi \in \bracsn{\pi(x)\eta|x \in A, \eta \in H}^\perp$, then for any $x \in A$ and $\eta \in H$, + \[ + \dpn{\pi(x)\xi, \eta}{H} = \dpn{\xi, \pi(x^*)\eta}{H} = 0 + \] + + so $\pi(x)\xi = 0$ for all $x \in A$. + + $\neg (2) \Rightarrow \neg (1)$: Let $\xi \in H \setminus \bracsn{0}$ with $\pi(x)\xi = 0$ for all $x \in A$, then for any $x \in A$ and $\eta \in H$, + \[ + \dpn{\pi(x)\eta, \xi}{H} = \dpn{\eta, \pi(x)\xi}{H} = 0 + \] + + so $\bracsn{\pi(x)\eta|x \in A, \eta \in H} \subset \bracsn{\xi}^\perp$, and cannot be dense in $H$. +\end{proof} + + \begin{proposition} \label{proposition:star-homomorphism-contractive} Let $A, B$ be unital $C^*$-algebras and $\phi: A \to B$ be a unital *-homomorphism, then for each $x \in A$,