First draft of representation for groups.
This commit is contained in:
Bokuan Li
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\section{The GNS Construction for Groups}
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\label{section:gns-lcg}
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\begin{definition}[Positive Linear Functional]
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\label{definition:group-l1-positive-functional}
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Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$\footnote{Due to technical difficulties, positive linear functionals are a priori not represented by $L^\infty$ functions. }, then $\phi$ is \textbf{positive} if $\dpn{f^* * f, \phi}{L^1(G; \complex)} \ge 0$ for all $f \in L^1(G)$.
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\end{definition}
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\begin{definition}[State]
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\label{definition:group-l1-state}
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Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$ be a positive linear functional, then $\phi$ is a \textbf{state} if $\norm{\phi}_{L^1(G; \complex)^*} = 1$. The set $S(G)$ of all states on $L^1(G; \complex)$ is the \textbf{state space} of $G$.
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\end{definition}
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\begin{definition}[GNS Triple (Locally Compact Group)]
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\label{definition:gns-triple-lcg}
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Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$ be a positive linear functional. For each $f, g \in L^1(G; \complex)$, let
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\[
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\dpn{f, g}{\phi} = \dpn{g^* * f, \phi}{L^1(G; \complex)}
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\]
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Let $N_\phi = \bracsn{f \in L^1(G; \complex)| \dpn{f, f}{\phi} = 0}$, $H_\phi^0 = L^1(G; \complex)/N_\phi$, $H_\phi$ be its completion with respect to $\dpn{\cdot, \cdot}{\phi}$, and
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\[
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\pi_\phi^0: G \to B(H_\phi^0) \quad \pi_\phi^0(x)(f + N_\phi) = L_xf + N_\phi
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\]
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For each $x \in G$, let $\pi_\phi(x)$ be the continuous extension of $\pi_\phi^0(x)$ to an element of $B(H_\phi)$, then:
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\begin{enumerate}
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\item $H_\phi$ equipped with the continuous extension of $\dpn{\cdot, \cdot}{\phi}$ is a Hilbert space.
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\item $(H_\phi, \pi_\phi)$ is a well-defined unitary representation of $G$.
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\end{enumerate}
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Let $\pi_\phi: L^1(G; \complex) \to B(H_\phi)$ also denote the *-representation of $L^1(G; \complex)$ determined by $\pi_\phi$, then:
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\begin{enumerate}[start=2]
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\item For each $f, g \in L^1(G; \complex)$, $\pi_\phi(f)(g + N_\phi) = f * g + N_\phi$.
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\item There exists a cyclic vector $\xi_\phi \in H_\phi$ for $\pi_\phi$ such that for each $f, g \in L^1(G; \complex)$,
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\[
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\dpn{f, g}{\phi} = \dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi}
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\]
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\item For each $x \in G$, let $\Phi(x) = \dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}$, then $\Phi \in BC(G; \complex)$ and
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\[
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\dpn{f, \phi}{L^1(G; \complex)} = \int_G f(x)\Phi(x)dx
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\]
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for all $f \in L^1(G; \complex)$.
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\end{enumerate}
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Moreover,
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\begin{enumerate}
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\item[(U)] Let $(K, \tau)$ be a unitary representation of $G$ with cyclic vector $\eta$ such that $\dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi} = \dpn{\tau(x)\eta, \eta}{K}$, then there exists a unitary equivalence $U: H_\phi \to K$ such that $U\xi_\phi = \eta$.
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\end{enumerate}
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The representation $(H_\phi, \pi_\phi)$ is the \textbf{unitary cyclic representation of $G$ induced by $\phi$}, and the triple $(H_\phi, \pi_\phi, \xi_\phi)$ is the \textbf{Gelfand-Naimark-Segal (GNS) triple associated with $\phi$}.
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\end{definition}
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\begin{proof}[Proof, {{\cite[Theorem 3.20]{FollandHarmonic}}}. ]
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(1): Since $\phi$ is positive, $\dpn{\cdot, \cdot}{\phi}$ is a pseudo inner product, and $H_\phi$ equipped with its continuous extension is a Hilbert space.
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(2): By the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}, for any $f, g \in L^1(G; \complex)$,
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\[
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\dpn{f, g}{\phi} \le \dpn{f, f}{\phi}^{1/2} \cdot \dpn{g, g}{\phi}^{1/2}
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\]
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In particular, if $g \in N_\phi$, then
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\begin{align*}
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\dpn{f * g, f * g}{\phi} &= \dpn{g^* * f^* * f * g, \phi}{L^1(G; \complex)} \\
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&= \dpn{g^** (f^**f*g), \phi}{L^1(G; \complex)} = \dpn{f^**f*g, g}{\phi}
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\end{align*}
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so $N_\phi$ is a left ideal of $L^1(G; \complex)$. Let $x \in G$, then $(L_xg)^* = L_{-x}g^*$, and
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\begin{align*}
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\dpn{L_xf, L_xg}{\phi} &= \dpn{(L_xg)^* * L_xf, \phi}{L^1(G; \complex)} = \dpn{(L_{-x}g^*) * (L_xf), \phi}{L^1(G; \complex)} \\
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&= \dpn{g^* * f, \phi}{L^1(G; \complex)} = \dpn{f, g}{\phi}
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\end{align*}
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by \autoref{proposition:convolution-function-gymnastics}. Therefore $\pi_\phi^0(x)$ extends to a unitary map on $H_\phi$, and $\pi_\phi$ is a well-defined unitary representation of $G$, which is strong-operator continuous by \hyperref[continuity of translation]{proposition:haar-translation}.
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(3): Let $f, g, h \in L^1(G; \complex)$, then after passing through \hyperref[Riemann sums]{lemma:convolution-distribute-functional},
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\begin{align*}
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\dpn{\pi_\phi(f)g + N_\phi, h + N_\phi}{H_\phi} &= \int_G f(x)\dpn{\pi_\phi(x)g + N_\phi, h + N_\phi}{H_\phi} dx \\
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&= \int_G f(x)\dpn{L_xg, h}{\phi} dx = \dpn{f * g, h}{\phi} \\
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&= \dpn{f * g + N_\phi, h + N_\phi}{H_\phi}
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\end{align*}
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As the above holds for all $h \in L^1(G; \complex)$, $\pi_\phi(f)(g + N_\phi) = f * g + N_\phi$.
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(4): Using \autoref{proposition:group-algebra-approximate-identity}, let $\angles{\psi_\alpha}_{\alpha \in A} \subset L^1(G; \complex)$ be an approximate identity, and $\xi_\phi \in H_\phi$ be defined by
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\[
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\dpn{\xi_\phi, g + N_\phi}{H_\phi} = \lim_{\alpha \in A}\dpn{\psi_\alpha, g}{\phi} \quad \forall g \in L^1(G; \complex)
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\]
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then $\normn{\xi_\phi}_{H_\phi} \le \norm{\phi}_{L^1(G; \complex)^*}^{1/2}$. Moreover, for any $f, g \in L^1(G; \complex)$,
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\begin{align*}
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\dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} &= \lim_{\alpha \in A}\dpn{\pi_\phi(f)(\psi_\alpha + N_\phi), \pi_\phi(g)\xi_\phi}{H_\phi} \\
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&= \lim_{\alpha \in A}\dpn{f * \psi_\alpha + N_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} \\
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&= \dpn{f + N_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} = \dpn{f + N_\phi, g + N_\phi}{H_\phi}
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\end{align*}
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\end{proof}
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\begin{corollary}
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\label{corollary:group-l1-positive-function}
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Let $G$ be a locally compact group, $\phi \in L^1(G; \complex)^*$ be a positive linear functional, $(H_\phi, \pi_\phi, \xi_\phi)$ be the GNS triple associated with $\phi$. Then:
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\begin{enumerate}
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\item For each $x \in G$, let $\Phi(x) = \dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}$, then $\Phi \in BC(G; \complex)$ with
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\[
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\dpn{f, \phi}{L^1(G; \complex)} = \int_G f(x)\Phi(x) dx
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\]
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for all $f \in L^1(G; \complex)$.
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\item $\Phi(1_G) = \norm{\Phi}_u = \norm{\phi}_{L^1(G; \complex)^*}$.
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\end{enumerate}
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\end{corollary}
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\begin{proof}
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(1): For any $f \in L^1(G; \complex)$
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\[
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\dpn{\pi_\phi(f)\xi_\phi, \xi_\phi}{H_\phi} = \int_G f(x)\dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}dx = \int_G f(x)\Phi(x)dx
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\]
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by definition of the representation. Since $\pi_\phi$ is strong-operator continuous, $\Phi \in C(G; \complex)$.
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(2): By \autoref{theorem:lp-dual-function}, $\Phi \in BC(G; \complex)$ with $\norm{\Phi}_u = \norm{\phi}_{L^1(G; \complex)^*}$. In addition, for any $x \in G$,
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\[
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|\Phi(x)| = |\dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}| \le \normn{\xi_\phi}_{H_\phi}^2 = \Phi(1_G)
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\]
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\end{proof}
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\begin{proposition}
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\label{proposition:gns-irreducible-lcg}
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Let $G$ be a locally compact group, $\phi \in S(G)$ be a state, and $(H_\phi, \pi_\phi, \xi_\phi)$ be the GNS triple associated with $\phi$, then $(H_\phi, \pi_\phi)$ is irreducible if and only if $\phi$ is an extreme point of $S(G)$.
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\end{proposition}
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\begin{proof}
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($\Rightarrow$): Suppose that $(H_\phi, \pi_\phi)$ is irreducible. Let $\psi_1, \psi_2 \in S(G)$ and $t \in (0, 1)$ such that $\phi = t\psi_1 + (1 - t)\psi_2$. Define
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\[
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T: H_\phi \to H_\phi \quad \dpn{T\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} = \dpn{f, g}{\psi_1} = \dpn{g^* * f, \psi_1}{L^1(G; \complex)}
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\]
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where $f, g \in L^1(G; \complex)$. Since both $\psi_1$ and $\psi_2$ are positive-definite functions, $\phi - t\psi_1$ is also positive-definite. In particular, $\dpn{f, f}{\psi_1} \le t^{-1}\dpn{f, f}{\phi}$ for all $f \in L^1(G; \complex)$, so $T$ extends continuously into a positive operator on $H_\phi$.
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For any $x \in G$ and $f, g \in L^1(G; \complex)$,
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\begin{align*}
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\dpn{\pi_\phi(x)T\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} &= \dpn{T\pi_\phi(f)\xi_\phi, \pi_\phi(x^{-1})\pi_\phi(g)\xi_\phi}{H_\phi} \\
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&= \dpn{T\pi_\phi(f)\xi_\phi, \pi_\phi(L_{x^{-1}}g)\xi_\phi}{H_\phi} \\
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&= \dpn{f, L_{x^{-1}}g}{\psi_1} = \dpn{(L_{x^{-1}}g)^* * f, \psi_1}{L^1(G; \complex)} \\
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&= \dpn{g^* * (L_xf), \psi_1}{L^1(G; \complex)} = \dpn{L_xf, g}{\psi_1} \\
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&= \dpn{T\pi_\phi(L_xf)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} \\
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&= \dpn{T\pi_\phi(x)\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi}
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\end{align*}
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so $T \in \pi_\phi(G)'$. By \hyperref[Schur's Lemma]{lemma:lcg-schur}, there exists $\lambda \in \complex \setminus \bracsn{0}$ such that $T = \lambda I$.
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Using \autoref{proposition:group-algebra-approximate-identity}, let $\angles{g_\alpha}_{\alpha \in A} \subset L^1(G; \complex)$ be an approximate identity for $L^1(G; \complex)$, then for any $f \in L^1(G; \complex)$,
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\[
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\dpn{f, \psi_1}{L^1(G; \complex)} = \lim_{\alpha \in A}\dpn{f, g_\alpha}{\psi_1} = \lim_{\alpha \in A}\lambda\dpn{f, g_\alpha}{\phi} = \lambda \dpn{f, \phi}{L^1(G; \complex)}
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\]
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Since $\phi, \psi_1 \in S(G)$, $\lambda = 1$, and $\phi = \psi_1$. As the argument is symmetric, $\phi = \psi_1 = \psi_2$.
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($\Leftarrow$): Suppose that $(H_\phi, \pi_\phi)$ is reducible. By \hyperref[Schur's Lemma]{lemma:lcg-schur}, $\pi_\phi(G)'$ is non-trivial. Since $\pi_\phi(G)'$ is a von Neumann algebra, it admits a non-trivial projection $P \in B(H_\phi)$ by \autoref{theorem:vn-projection-norm-dense}.
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Let $\xi_1 = P\xi_\phi/\normn{P\xi_\phi}_{H_\phi}$ and $\xi_2 = (1 - P)\xi_\phi/\normn{\xi_\phi - P\xi_\phi}_{H_\phi}$. For each $f \in L^1(G; \complex)$, let
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\[
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\dpn{f, \psi_1}{L^1(G; \complex)} = \dpn{\pi_\phi(f)\xi_1, \xi_1}{H_\phi} \quad
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\dpn{f, \psi_2}{L^1(G; \complex)} = \dpn{\pi_\phi(f)\xi_2, \xi_2}{H_\phi}
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\]
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then $\psi_1, \psi_2 \in S(G)$. Since $P \in \pi_\phi(G)'$, for any $f \in L^1(G; \complex)$,
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\begin{align*}
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\dpn{f, \phi}{L^1(G; \complex)} &= \dpn{\pi_\phi(f)\xi_\phi, \xi_\phi}{H_\phi} \\
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&= \dpn{\pi_\phi(f)P\xi_\phi, P\xi_\phi}{H_\phi} + \dpn{\pi_\phi(f)(I - P)\xi_\phi, (I - P)\xi_\phi}{H_\phi} \\
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&= \normn{P\xi_\phi}_{H_\phi}^2\dpn{\pi_\phi(f)\xi_1, \xi_1}{H_\phi} + \normn{(1 - P)\xi_\phi}_{H_\phi}^2 \dpn{\pi_\phi(f)\xi_2, \xi_2}{H_\phi} \\
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&= \normn{P\xi_\phi}_{H_\phi}^2 \dpn{f, \psi_1}{L^1(G; \complex)} + \normn{(1 - P)\xi_\phi}_{H_\phi}^2 \dpn{f, \psi_2}{L^1(G; \complex)}
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\end{align*}
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so $\phi$ is a strict convex combination of two states. If $\psi_1 = \phi$, then for any $f, g \in L^1(G; \complex)$,
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\begin{align*}
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\dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} &= \normn{P\xi_\phi}_{H_\phi}^{-2} \dpn{P\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} \\
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\dpn{[\pi_\phi(f) - \normn{P\xi_\phi}_{H_\phi}^{-2}P\pi_\phi(f)]\xi_\phi,\pi_\phi(g) \xi_\phi}{H_\phi} &= 0
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\end{align*}
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so $I - \normn{P\xi_\phi}_{H_\phi}^{-2}P = 0$, which contradicts the fact that $P$ is a non-trivial projection.
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\end{proof}
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@@ -3,4 +3,6 @@
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\input{./unit.tex}
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\input{./l1.tex}
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\input{./gns.tex}
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\input{./pos.tex}
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+165
-161
@@ -1,124 +1,6 @@
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\section{Positive-Definite Functions}
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\label{section:positive-definite-function}
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\begin{definition}[Positive Linear Functional]
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\label{definition:group-l1-positive-functional}
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Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$\footnote{Due to technical difficulties, positive linear functionals are a priori not represented by $L^\infty$ functions. }, then $\phi$ is \textbf{positive} if $\dpn{f^* * f, \phi}{L^1(G; \complex)} \ge 0$ for all $f \in L^1(G)$.
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\end{definition}
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\begin{definition}[GNS Triple (Locally Compact Group)]
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\label{definition:gns-triple-lcg}
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Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$. For each $f, g \in L^1(G; \complex)$, let
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\[
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\dpn{f, g}{\phi} = \dpn{g^* * f, \phi}{L^1(G; \complex)}
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\]
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Let $N_\phi = \bracsn{f \in L^1(G; \complex)| \dpn{f, f}{\phi} = 0}$, $H_\phi^0 = L^1(G; \complex)/N_\phi$, $H_\phi$ be its completion with respect to $\dpn{\cdot, \cdot}{\phi}$, and
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\[
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\pi_\phi^0: G \to B(H_\phi^0) \quad \pi_\phi^0(x)(f + N_\phi) = L_xf + N_\phi
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\]
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For each $x \in G$, let $\pi_\phi(x)$ be the continuous extension of $\pi_\phi^0(x)$ to an element of $B(H_\phi)$, then:
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\begin{enumerate}
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\item $H_\phi$ equipped with the continuous extension of $\dpn{\cdot, \cdot}{\phi}$ is a Hilbert space.
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\item $(H_\phi, \pi_\phi)$ is a well-defined unitary representation of $G$.
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\end{enumerate}
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Let $\pi_\phi: L^1(G; \complex) \to B(H_\phi)$ also denote the *-representation of $L^1(G; \complex)$ determined by $\pi_\phi$, then:
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\begin{enumerate}[start=2]
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\item For each $f, g \in L^1(G; \complex)$, $\pi_\phi(f)(g + N_\phi) = f * g + N_\phi$.
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\item There exists a cyclic vector $\xi_\phi \in H_\phi$ for $\pi_\phi$ such that for each $f, g \in L^1(G; \complex)$,
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\[
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\dpn{f, g}{\phi} = \dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi}
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\]
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\item For each $x \in G$, let $\Phi(x) = \dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}$, then $\Phi \in BC(G; \complex)$ and
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\[
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\dpn{f, \phi}{L^1(G; \complex)} = \int_G f(x)\Phi(x)dx
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\]
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for all $f \in L^1(G; \complex)$.
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\end{enumerate}
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Moreover,
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\begin{enumerate}
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\item[(U)] Let $(K, \tau)$ be a unitary representation of $G$ with cyclic vector $\eta$ such that $\dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi} = \dpn{\tau(x)\eta, \eta}{K}$, then there exists a unitary equivalence $U: H_\phi \to K$ such that $U\xi_\phi = \eta$.
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\end{enumerate}
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The representation $(H_\phi, \pi_\phi)$ is the \textbf{unitary cyclic representation of $G$ induced by $\phi$}, and the triple $(H_\phi, \pi_\phi, \xi_\phi)$ is the \textbf{Gelfand-Naimark-Segal (GNS) triple associated with $\phi$}.
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\end{definition}
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\begin{proof}[Proof, {{\cite[Theorem 3.20]{FollandHarmonic}}}. ]
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(1): Since $\phi$ is positive-definite, $\dpn{\cdot, \cdot}{\phi}$ is a pseudo inner product, and $H_\phi$ equipped with its continuous extension is a Hilbert space.
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(2): By the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}, for any $f, g \in L^1(G; \complex)$,
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\[
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\dpn{f, g}{\phi} \le \dpn{f, f}{\phi}^{1/2} \cdot \dpn{g, g}{\phi}^{1/2}
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\]
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In particular, if $g \in N_\phi$, then
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\begin{align*}
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\dpn{f * g, f * g}{\phi} &= \dpn{g^* * f^* * f * g, \phi}{L^1(G; \complex)} \\
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&= \dpn{g^** (f^**f*g), \phi}{L^1(G; \complex)} = \dpn{f^**f*g, g}{\phi}
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\end{align*}
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so $N_\phi$ is a left ideal of $L^1(G; \complex)$. Let $x \in G$, then $(L_xg)^* = L_{-x}g^*$, and
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\begin{align*}
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\dpn{L_xf, L_xg}{\phi} &= \dpn{(L_xg)^* * L_xf, \phi}{L^1(G; \complex)} = \dpn{(L_{-x}g^*) * (L_xf), \phi}{L^1(G; \complex)} \\
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&= \dpn{g^* * f, \phi}{L^1(G; \complex)} = \dpn{f, g}{\phi}
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\end{align*}
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|
||||
by \autoref{proposition:convolution-function-gymnastics}. Therefore $\pi_\phi^0(x)$ extends to a unitary map on $H_\phi$, and $\pi_\phi$ is a well-defined unitary representation of $G$, which is strong-operator continuous by \hyperref[continuity of translation]{proposition:haar-translation}.
|
||||
|
||||
(3): Let $f, g, h \in L^1(G; \complex)$, then after passing through \hyperref[Riemann sums]{lemma:convolution-distribute-functional},
|
||||
\begin{align*}
|
||||
\dpn{\pi_\phi(f)g + N_\phi, h + N_\phi}{H_\phi} &= \int_G f(x)\dpn{\pi_\phi(x)g + N_\phi, h + N_\phi}{H_\phi} dx \\
|
||||
&= \int_G f(x)\dpn{L_xg, h}{\phi} dx = \dpn{f * g, h}{\phi} \\
|
||||
&= \dpn{f * g + N_\phi, h + N_\phi}{H_\phi}
|
||||
\end{align*}
|
||||
|
||||
As the above holds for all $h \in L^1(G; \complex)$, $\pi_\phi(f)(g + N_\phi) = f * g + N_\phi$.
|
||||
|
||||
(4): Using \autoref{proposition:group-algebra-approximate-identity}, let $\angles{\psi_\alpha}_{\alpha \in A} \subset L^1(G; \complex)$ be an approximate identity, and $\xi_\phi \in H_\phi$ be defined by
|
||||
\[
|
||||
\dpn{\xi_\phi, g + N_\phi}{H_\phi} = \lim_{\alpha \in A}\dpn{\psi_\alpha, g}{\phi} \quad \forall g \in L^1(G; \complex)
|
||||
\]
|
||||
|
||||
then $\normn{\xi_\phi}_{H_\phi} \le \norm{\phi}_{L^1(G; \complex)^*}^{1/2}$. Moreover, for any $f, g \in L^1(G; \complex)$,
|
||||
\begin{align*}
|
||||
\dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} &= \lim_{\alpha \in A}\dpn{\pi_\phi(f)(\psi_\alpha + N_\phi), \pi_\phi(g)\xi_\phi}{H_\phi} \\
|
||||
&= \lim_{\alpha \in A}\dpn{f * \psi_\alpha + N_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} \\
|
||||
&= \dpn{f + N_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} = \dpn{f + N_\phi, g + N_\phi}{H_\phi}
|
||||
\end{align*}
|
||||
\end{proof}
|
||||
|
||||
\begin{corollary}
|
||||
\label{corollary:group-l1-positive-function}
|
||||
Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$ be a positive linear functional, then:
|
||||
\begin{enumerate}
|
||||
\item There exists $\Phi \in BC(G; \complex)$ such that
|
||||
\[
|
||||
\dpn{f, \phi}{L^1(G; \complex)} = \int_G f(x)\Phi(x) dx
|
||||
\]
|
||||
|
||||
for all $f \in L^1(G; \complex)$.
|
||||
\item $\Phi(1_G) = \norm{\Phi}_u = \norm{\phi}_{L^1(G; \complex)^*}$.
|
||||
\end{enumerate}
|
||||
\end{corollary}
|
||||
\begin{proof}
|
||||
(1): Let $(H_\phi, \pi_\phi, \xi_\phi)$ be the \hyperref[GNS triple]{definition:gns-triple-lcg} associated with $\phi$. For each $x \in G$, let $\Phi(x) = \dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H}$, then for any $f \in L^1(G; \complex)$
|
||||
\[
|
||||
\dpn{\pi_\phi(f)\xi_\phi, \xi_\phi}{H_\phi} = \int_G f(x)\dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}dx = \int_G f(x)\Phi(x)dx
|
||||
\]
|
||||
|
||||
by definition of the representation. Since $\pi_\phi$ is strong-operator continuous, $\Phi \in C(G; \complex)$.
|
||||
|
||||
(2): By \autoref{theorem:lp-dual-function}, $\Phi \in BC(G; \complex)$ with $\norm{\Phi}_u = \norm{\phi}_{L^1(G; \complex)^*}$. In addition, for any $x \in G$,
|
||||
\[
|
||||
|\Phi(x)| = |\dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}| \le \normn{\xi_\phi}_{H_\phi}^2 = \Phi(1_G)
|
||||
\]
|
||||
\end{proof}
|
||||
|
||||
\begin{lemma}
|
||||
\label{lemma:group-positive-computation-gymnastics}
|
||||
@@ -129,15 +11,12 @@
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
\begin{align*}
|
||||
\int_G (f^* * f)(x)\phi(x) dx &= \iint_G \Delta_G(y^{-1})\ol{f(y^{-1})}f(y^{-1}x)\phi(x)dydx \\
|
||||
\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} f(x) \ol{f(y)} \phi(y^{-1}x)dydx
|
||||
\end{align*}
|
||||
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
||||
\begin{definition}[Positive-Definite Function]
|
||||
\label{definition:group-positive-definite-function}
|
||||
Let $G$ be a locally compact group and $\phi \in BC(G; \complex)$, then the following are equivalent:
|
||||
@@ -149,14 +28,16 @@
|
||||
\]
|
||||
\end{enumerate}
|
||||
|
||||
If the above holds, then $\phi$ is \textbf{positive-definite}. The set $P(G)$ is the \textbf{space of positive-definite functions} on $G$, $P_0(G) = \bracsn{\phi \in P(G)|\norm{\phi}_u \le 1}$, and $S(G) = \bracsn{\phi \in P(G)|\norm{\phi}_u = 1}$.
|
||||
If the above holds, then $\phi$ is \textbf{positive-definite}.
|
||||
|
||||
The set $P(G)$ is the \textbf{space of positive-definite functions} on $G$. The set $P_0(G)$ is the space of positive-definite functions of norm at most 1. Finally, the set $S(G)$ of positive-definite functions of norm 1 is the \textbf{state space} of $G$.
|
||||
\end{definition}
|
||||
\begin{proof}[Proof, {{\cite[Proposition 3.35]{FollandHarmonic}}}. ]
|
||||
(1) $\Rightarrow$ (2): Using \autoref{proposition:group-algebra-approximate-identity}, let $\angles{\psi_\alpha}_{\alpha \in A} \subset L^1(G; \complex)$ be an approximate identity such that for all $U \in \cn_G(1)$, there exists $\alpha_0 \in A$ with $\supp{\psi_\alpha} \subset U$ for all $\alpha \ge \alpha_0$.
|
||||
|
||||
For each $\alpha \in A$, $\seqf{\lambda_j} \subset \complex$, and $\seqf{x_j} \subset G$, let $f_\alpha = \sum_{j = 1}^n \lambda_j L_{x_j}\psi_\alpha$, then by continuity of $\phi$,
|
||||
\begin{align*}
|
||||
0 \le \int_G (f^*_\alpha * f_\alpha) \phi &= \sum_{i, j = 1}^n \lambda_i \ol{\lambda_j} \iint_{G \times G} \psi_\alpha(x_i^{-1}y)\ol{\psi_\alpha(x_j^{-1}y)}\phi(z^{-1}y)dydz \\
|
||||
0 \le \int_G (f^*_\alpha * f_\alpha) \phi &= \sum_{i, j = 1}^n \lambda_i \ol{\lambda_j} \iint_{G \times G} \psi_\alpha(x_i^{-1}y)\ol{\psi_\alpha(x_j^{-1}z)}\phi(z^{-1}y)dydz \\
|
||||
0 \le \lim_{\alpha \in A}\int_G (f^*_\alpha * f_\alpha) \phi &= \sum_{i, j = 1}^n \lambda_i \ol{\lambda_j}\phi(x_j^{-1}x_i)
|
||||
\end{align*}
|
||||
|
||||
@@ -174,60 +55,183 @@
|
||||
and the above holds for all $\eps > 0$, $\int_G (f^* * f) \phi \ge 0$. By \hyperref[density of $C_c(G; \complex)$ in $L^1(G; \complex)$]{proposition:radon-cc-dense} and continuity of the convolution product, $\int_G (g^* * g) \phi \ge 0$ for all $g \in L^1(G; \complex)$.
|
||||
\end{proof}
|
||||
|
||||
\begin{proposition}
|
||||
\label{proposition:gns-irreducible-lcg}
|
||||
Let $G$ be a locally compact group, $\phi \in S(G)$ be a state, and $(H_\phi, \pi_\phi, \xi_\phi)$ be the GNS triple associated with $\phi$, then $(H_\phi, \pi_\phi)$ is irreducible if and only if $\phi$ is an extreme point of $S(G)$.
|
||||
\end{proposition}
|
||||
\begin{lemma}
|
||||
\label{lemma:positive-definite-function-conjugate}
|
||||
Let $G$ be a locally compact group and $\phi \in BC(G; \complex)$ be a positive-definite function, then $\ol \phi$ is also positive definite.
|
||||
\end{lemma}
|
||||
\begin{proof}[Proof, {{\cite[Proposition 3.14]{FollandHarmonic}}}. ]
|
||||
For any $f \in L^1(G; \complex)$,
|
||||
\[
|
||||
\int_G (f^* * f)\ol \phi = \int_G (\ol{f}^* * \ol{f}) \phi
|
||||
\]
|
||||
\end{proof}
|
||||
|
||||
\begin{lemma}
|
||||
\label{lemma:positive-definite-from-representation}
|
||||
Let $G$ be a locally compact group, then:
|
||||
\begin{enumerate}
|
||||
\item For any unitary representation $(H, \pi)$ of $G$ and $\xi \in H$, $\phi(x) = \dpn{\pi(x)\xi, \xi}{H}$ is a positive-definite function.
|
||||
\item For any $f \in L^2(G; \complex)$, let $\tilde f(x) = \ol{f(x^{-1})}$, then $f * \tilde f$ is a positive-definite function.
|
||||
\end{enumerate}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
($\Rightarrow$): Suppose that $(H_\phi, \pi_\phi)$ is irreducible. Let $\psi_1, \psi_2 \in S(G)$ and $t \in (0, 1)$ such that $\phi = t\psi_1 + (1 - t)\psi_2$. Define
|
||||
\[
|
||||
T: H_\phi \to H_\phi \quad \dpn{T\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} = \dpn{f, g}{\psi_1} = \dpn{g^* * f, \psi_1}{L^1(G; \complex)}
|
||||
\]
|
||||
|
||||
where $f, g \in L^1(G; \complex)$. Since both $\psi_1$ and $\psi_2$ are positive-definite functions, $\phi - t\psi_1$ is also positive-definite. In particular, $\dpn{f, f}{\psi_1} \le t^{-1}\dpn{f, f}{\phi}$ for all $f \in L^1(G; \complex)$, so $T$ extends continuously into a positive operator on $H_\phi$.
|
||||
|
||||
For any $x \in G$ and $f, g \in L^1(G; \complex)$,
|
||||
(1): Let $f \in L^1(G; \complex)$, then by \autoref{lemma:group-positive-computation-gymnastics},
|
||||
\begin{align*}
|
||||
\dpn{\pi_\phi(x)T\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} &= \dpn{T\pi_\phi(f)\xi_\phi, \pi_\phi(x^{-1})\pi_\phi(g)\xi_\phi}{H_\phi} \\
|
||||
&= \dpn{T\pi_\phi(f)\xi_\phi, \pi_\phi(L_{x^{-1}}g)\xi_\phi}{H_\phi} \\
|
||||
&= \dpn{f, L_{x^{-1}}g}{\psi_1} = \dpn{(L_{x^{-1}}g)^* * f, \psi_1}{L^1(G; \complex)} \\
|
||||
&= \dpn{g^* * (L_xf), \psi_1}{L^1(G; \complex)} = \dpn{L_xf, g}{\psi_1} \\
|
||||
&= \dpn{T\pi_\phi(L_xf)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} \\
|
||||
&= \dpn{T\pi_\phi(x)\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi}
|
||||
\int_G (f^* * f)(x)\phi(x)dx &= \iint_{G \times G} f(x)\ol{f(y)}\phi(y^{-1}x)dxdy \\
|
||||
&= \iint_G \dpn{f(x)\pi(x)\xi, f(y)\pi(y)\xi}{H}dxdy = \norm{\pi(f)\xi}_H^2 \ge 0
|
||||
\end{align*}
|
||||
|
||||
so $T \in \pi_\phi(G)'$. By \hyperref[Schur's Lemma]{lemma:lcg-schur}, there exists $\lambda \in \complex \setminus \bracsn{0}$ such that $T = \lambda I$.
|
||||
|
||||
Using \autoref{proposition:group-algebra-approximate-identity}, let $\angles{g_\alpha}_{\alpha \in A} \subset L^1(G; \complex)$ be an approximate identity for $L^1(G; \complex)$, then for any $f \in L^1(G; \complex)$,
|
||||
(2): Let $\pi: G \to B(L^2(G; \complex))$ be the left regular representation of $G$, then
|
||||
\[
|
||||
\dpn{f, \psi_1}{L^1(G; \complex)} = \lim_{\alpha \in A}\dpn{f, g_\alpha}{\psi_1} = \lim_{\alpha \in A}\lambda\dpn{f, g_\alpha}{\phi} = \lambda \dpn{f, \phi}{L^1(G; \complex)}
|
||||
\dpn{\pi(x)f, f}{L^2(G; \complex)} = \int_G f(x^{-1}y)\ol{f(y)}dy = \ol{(f * \tilde f)(x)}
|
||||
\]
|
||||
|
||||
Since $\phi, \psi_1 \in S(G)$, $\lambda = 1$, and $\phi = \psi_1$. As the argument is symmetric, $\phi = \psi_1 = \psi_2$.
|
||||
By (1) and \autoref{lemma:positive-definite-function-conjugate}, $f * \tilde f$ is positive-definite.
|
||||
\end{proof}
|
||||
|
||||
($\Leftarrow$): Suppose that $(H_\phi, \pi_\phi)$ is reducible. By \hyperref[Schur's Lemma]{lemma:lcg-schur}, $\pi_\phi(G)'$ is non-trivial. Since $\pi_\phi(G)'$ is a von Neumann algebra, it admits a non-trivial projection $P \in B(H_\phi)$ by \autoref{theorem:vn-projection-norm-dense}.
|
||||
|
||||
Let $\xi_1 = P\xi_\phi/\normn{P\xi_\phi}_{H_\phi}$ and $\xi_2 = (1 - P)\xi_\phi/\normn{\xi_\phi - P\xi_\phi}_{H_\phi}$. For each $f \in L^1(G; \complex)$, let
|
||||
\begin{lemma}
|
||||
\label{lemma:positive-definite-cc-decompose}
|
||||
Let $G$ be a locally compact group, then for any $f * g \in C_c(G; \complex) * C_c(G; \complex)$, there exists $\bracsn{h_k}_0^3 \subset C_c(G; \complex) \cap P(G)$ such that
|
||||
\[
|
||||
\dpn{f, \psi_1}{L^1(G; \complex)} = \dpn{\pi_\phi(f)\xi_1, \xi_1}{H_\phi} \quad
|
||||
\dpn{f, \psi_2}{L^1(G; \complex)} = \dpn{\pi_\phi(f)\xi_2, \xi_2}{H_\phi}
|
||||
f * g = \sum_{k = 0}^3 i^k h_k
|
||||
\]
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
For each $h \in C_c(G; \complex)$, denote $\tilde h(x) = \ol{h(x^{-1})}$, then by \hyperref[polarisation]{proposition:polarisation-complex},
|
||||
\[
|
||||
f * \tilde g = \frac{1}{4}\sum_{k = 0}^3 i^k (f + i^k g) * (f + i^k g)^\sim
|
||||
\]
|
||||
|
||||
then $\psi_1, \psi_2 \in S(G)$. Since $P \in \pi_\phi(G)'$, for any $f \in L^1(G; \complex)$,
|
||||
Since $C_c(G; \complex) \subset L^2(G; \complex)$, (2) of \autoref{lemma:positive-definite-from-representation} implies that $(f + i^k g) * (f + i^k g)^\sim$ is positive-definite.
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
||||
|
||||
\begin{lemma}
|
||||
\label{lemma:lcg-positive-definite-function-concentration}
|
||||
Let $G$ be a locally compact group and $\phi \in S(G)$, then for each $x, y \in G$,
|
||||
\[
|
||||
|\phi(x) - \phi(y)|^2 \le 2 - 2\text{Re}\phi(yx^{-1})
|
||||
\]
|
||||
\end{lemma}
|
||||
\begin{proof}[Proof, {{\cite[Lemma 3.30]{FollandHarmonic}}}. ]
|
||||
Let $(H_\phi, \pi_\phi, \xi_\phi)$ be the \hyperref[GNS triple]{definition:gns-triple-lcg} associated with $\phi$, then by \autoref{corollary:group-l1-positive-function}, $\phi(x) = \dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}$ for all $x \in G$. Since $\norm{\phi}_u = \phi(1_G) = 1$, $\norm{\xi_\phi}_{H_\phi} = 1$. As such,
|
||||
\begin{align*}
|
||||
|\phi(x) - \phi(y)|^2 &= |\dpn{[\pi_\phi(x) - \pi_\phi(y)]\xi_\phi, \xi_\phi}{H_\phi}|^2 \\
|
||||
&= |\dpn{\xi_\phi, [\pi_\phi(x^{-1}) - \pi_\phi(y^{-1})]\xi_\phi}{H_\phi}|^2 \\
|
||||
&\le \normn{[\pi_\phi(x^{-1}) - \pi_\phi(y^{-1})]\xi_\phi}_{H_\phi}^2 \\
|
||||
&= 2 - 2\text{Re}\dpn{\pi_\phi(x^{-1})\xi_\phi, \pi_\phi(y^{-1})\xi_\phi}{H_\phi} \\
|
||||
&= 2 - 2\text{Re}\dpn{\pi_\phi(yx^{-1})\xi_\phi, \xi_\phi}{H_\phi} = 2 - 2\text{Re}\phi(yx^{-1})
|
||||
\end{align*}
|
||||
\end{proof}
|
||||
|
||||
|
||||
\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 $L^1(G; \complex)^*$.
|
||||
\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 $\phi, \psi \in \mathcal{F}$,
|
||||
\begin{align*}
|
||||
\dpn{f, \phi}{L^1(G; \complex)} &= \dpn{\pi_\phi(f)\xi_\phi, \xi_\phi}{H_\phi} \\
|
||||
&= \dpn{\pi_\phi(f)P\xi_\phi, P\xi_\phi}{H_\phi} + \dpn{\pi_\phi(f)(I - P)\xi_\phi, (I - P)\xi_\phi}{H_\phi} \\
|
||||
&= \normn{P\xi_\phi}_{H_\phi}^2\dpn{\pi_\phi(f)\xi_1, \xi_1}{H} + \normn{(1 - P)\xi_\phi}_{H_\phi}^2 \dpn{\pi_\phi(f)\xi_2, \xi_2}{H} \\
|
||||
&= \normn{P\xi_\phi}_{H_\phi}^2 \dpn{f, \psi_1}{A} + \normn{(1 - P)\xi_\phi}_{H_\phi}^2 \dpn{f, \psi_2}{L^1(G; \complex)}
|
||||
\max_{1 \le j \le n}\abs{\int_{G} f_j (\phi - \psi)} &\le \max_{1 \le j \le n}\int_G|f_j(\phi - \psi)| \\
|
||||
&\le \max_{1 \le j \le n}\braks{\int_{K} |f_j(\phi - \psi)| + \int_{G \setminus K}|f_j(\phi - \psi)|} \\
|
||||
&\le \sup_{x \in K}|\phi(x) - \psi(x)|\max_{1 \le j \le n}\norm{f_j}_{L^1(G; \complex)} + \eps
|
||||
\end{align*}
|
||||
|
||||
so $\phi$ is a strict convex combination of two states. If $\psi_1 = \phi$, then for any $f, g \in L^1(G; \complex)$,
|
||||
($\subset$): Let $\phi \in S(G)$ and $V \in \cn_G(1_G)$ be relatively compact, then for each $x \in G$,
|
||||
\begin{align*}
|
||||
\dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} &= \normn{P\xi_\phi}_{H_\phi}^{-2} \dpn{P\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} \\
|
||||
\dpn{[\pi_\phi(f) - \normn{P\xi_\phi}_{H_\phi}^{-2}P\pi_\phi(f)]\xi_\phi,\pi_\phi(g) \xi_\phi}{H_\phi} &= 0
|
||||
\abs{\phi(x) - \frac{\one_V}{|V|} * \phi(x)} &\le \frac{1}{|V|}\int_V |\phi(x) - \phi(y^{-1}x)|dx \\
|
||||
&\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}
|
||||
\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 $x \in G$,
|
||||
\begin{align*}
|
||||
|\phi(x) - \psi(x)| &\le \abs{\phi(x) - \frac{\one_V}{|V|} * \phi(x)} + \frac{1}{|V|}\abs{\one_V * (\phi - \psi)(x)}\\
|
||||
&+ \abs{\psi(x) - \frac{\one_V}{|V|} * \psi(x)} \\
|
||||
&\le |V|^{1/2}\braks{\int_V 2 - 2\text{Re}(\phi(y)) dy}^{1/2} + \abs{\angles{\frac{L_x\one_V}{|V|}, \phi - \psi}_{L^1(G; \complex)}} \\
|
||||
&+ |V|^{1/2}\braks{\int_V 2 - 2\text{Re}(\psi(y)) dy}^{1/2} \\
|
||||
&\le 2|V|^{1/2}\braks{\int_V 2 - 2\text{Re}(\phi(y)) dy}^{1/2}+ |2\dpn{\one_V, \phi - \psi}{L^1(G; \complex)}|^{1/2} \\
|
||||
&+ \abs{\angles{\frac{L_x\one_V}{|V|}, \phi - \psi}_{L^1(G; \complex)}} \\
|
||||
&\le 4|V|\sup_{y \in V}|\phi(y) - 1|^{1/2}+ |2\dpn{\one_V, \phi - \psi}{L^1(G; \complex)}|^{1/2}\\
|
||||
&+ \abs{\angles{\frac{L_x\one_V}{|V|}, \phi - \psi}_{L^1(G; \complex)}} \\
|
||||
\end{align*}
|
||||
|
||||
For any $K \subset G$ compact,
|
||||
\begin{align*}
|
||||
\sup_{x \in K}|\phi(x) - \psi(x)| &\le 4|V|\sup_{y \in V}|\phi(y) - 1|^{1/2}+ |2\dpn{\one_V, \phi - \psi}{L^1(G; \complex)}|^{1/2} \\
|
||||
&+\sup_{x \in K}\abs{\angles{\frac{L_x\one_V}{|V|}, \phi - \psi}_{L^1(G; \complex)}}
|
||||
\end{align*}
|
||||
|
||||
Let $\eps > 0$. By \autoref{corollary:group-l1-positive-function}, $\phi$ is continuous with $\phi(1_G) = 1$, so there exists a relatively compact neighbourhood $V \in \cn_G(1_G)$ with $4|V|\sup_{y \in V}|\phi(y) - 1|^{1/2} < \eps$.
|
||||
|
||||
Since $S(G)$ is a bounded subset of $L^1(G; \complex)^*$, it is equicontinuous. By the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, the weak* topology and the topology of uniform convergence on compact subsets of $L^1(G; \complex)$ coincide on $S(G)$. In particular, given that $K$ is compact, $\bracsn{L_x\one_V|x \in K}$ is compact in $L^1(G; \complex)$, so there exists $W \in \cn_{S(G)}(\phi)$ such that
|
||||
\[
|
||||
|2\dpn{\one_V, \phi - \psi}{L^1(G; \complex)}|^{1/2} +\sup_{x \in K}\abs{\angles{\frac{L_x\one_V}{|V|}, \phi - \psi}_{L^1(G; \complex)}} \le \eps
|
||||
\]
|
||||
|
||||
for all $\psi \in W$. Therefore for any $\psi \in W$, $\sup_{x \in K}|\phi(x) - \psi(x)| \le 2\eps$.
|
||||
|
||||
|
||||
\end{proof}
|
||||
|
||||
|
||||
\begin{theorem}
|
||||
\label{theorem:lcg-state-extreme}
|
||||
Let $G$ be a locally compact group, then:
|
||||
\begin{enumerate}
|
||||
\item The extreme points of $P_1(G)$ is 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}
|
||||
\begin{proof}[Proof, {{\cite[Lemma 3.26, Theorem 3.27]{FollandHarmonic}}}. ]
|
||||
(1): Let $\phi \in S(G)$, $\psi_1, \psi_2 \in P_1(G)$, and $t \in (0, 1)$ such that $\phi = (1 - t)\psi_1 + t\psi_2$, then
|
||||
\[
|
||||
1 = \norm{\phi}_{L^1(G; \complex)^*} = (1 - t)\norm{\psi_1}_{L^1(G; \complex)^*} + t\norm{\psi_2}_{L^1(G; \complex)^*} \le 1
|
||||
\]
|
||||
|
||||
so $\norm{\psi_2}_{L^1(G; \complex)^*} = \norm{\psi_2}_{L^1(G; \complex)^*} = 1$, and $S(G)$ is an extreme subset of $P_1(G)$. By \autoref{lemma:extreme-subset-extreme-point}, every extreme point of $S(G)$ is an extreme point of $P_1(G)$.
|
||||
|
||||
On the other hand, every extreme point of $P_1(G)$ in $S(G)$ is an extreme point of $S(G)$.
|
||||
|
||||
It remains to show that $0$ is the only other extreme point of $P_1(G)$. To this end, let $\phi, \psi \in P_1(G)$ 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 $P_1(G)$. For any $\phi \in P_1(G) \setminus (S(G) \cup \bracsn{0})$, $\phi/\phi(1_G) \in S(G)$, and $\phi = 0 + \phi(1_G) \cdot \phi/\phi(1_G)$. Therefore $0$ is the only other extreme point of $P_1(G)$.
|
||||
|
||||
(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 $\bracsn{\psi \in P_1(G)|\ \norm{\psi}_{L^1(G; \complex)^*} > 1 - \eps}$ 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
|
||||
\[
|
||||
\max_{f \in \cf}\angles{f, \phi - \sum_{j = 1}^n \lambda_j \phi_j}_{L^1(G; \complex)} < \eps
|
||||
\]
|
||||
|
||||
In which case, $\frac{1}{1 - \lambda_0}\sum_{j = 1}^n \lambda_j = 1$ and
|
||||
\[
|
||||
\max_{f \in \cf}\angles{f, \sum_{j = 1}^n \lambda_j \phi_j - \frac{1}{1 - \lambda_0}\sum_{j = 1}^n \lambda_j \phi_j}_{L^1(G; \complex)} \le |\lambda_0| \max_{f \in \cf}\norm{f}_{L^1(G; \complex)} \le \eps
|
||||
\]
|
||||
|
||||
so
|
||||
\[
|
||||
\max_{f \in \cf}\angles{f, \phi - \sum_{j = 1}^n \frac{\lambda_j}{1 - \lambda_0} \phi_j}_{L^1(G; \complex)} < 2\eps
|
||||
\]
|
||||
\end{proof}
|
||||
|
||||
|
||||
\begin{theorem}[Gelfand-Raikov]
|
||||
\label{theorem:gelfand-raikov-lcg}
|
||||
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},
|
||||
\[
|
||||
\phi(x) = \dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi} =
|
||||
\dpn{\pi_\phi(y)\xi_\phi, \xi_\phi}{H_\phi} = \phi(y)
|
||||
\]
|
||||
|
||||
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 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 $g \in C_c(G; \complex)$.
|
||||
|
||||
so $I - \normn{P\xi_\phi}_{H_\phi}^{-2}P = 0$, which contradicts the fact that $P$ is a non-trivial projection.
|
||||
Should $x \ne y$, then \hyperref[Urysohn's Lemma]{lemma:lch-urysohn} implies that there exists $g \in C_c(G; \complex)$ with $g(x) \ne g(y)$. Since this is not the case, $x = y$.
|
||||
\end{proof}
|
||||
|
||||
|
||||
|
||||
Reference in New Issue
Block a user