diff --git a/src/harmonic/rep/gns.tex b/src/harmonic/rep/gns.tex new file mode 100644 index 0000000..c1b6a3d --- /dev/null +++ b/src/harmonic/rep/gns.tex @@ -0,0 +1,182 @@ +\section{The GNS Construction for Groups} +\label{section:gns-lcg} + +\begin{definition}[Positive Linear Functional] +\label{definition:group-l1-positive-functional} + 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 $f \in L^1(G)$. +\end{definition} + +\begin{definition}[State] +\label{definition:group-l1-state} + 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 $G$. +\end{definition} + +\begin{definition}[GNS Triple (Locally Compact Group)] +\label{definition:gns-triple-lcg} + 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 + \[ + \dpn{f, g}{\phi} = \dpn{g^* * f, \phi}{L^1(G; \complex)} + \] + + 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 + \[ + \pi_\phi^0: G \to B(H_\phi^0) \quad \pi_\phi^0(x)(f + N_\phi) = L_xf + N_\phi + \] + + 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: + \begin{enumerate} + \item $H_\phi$ equipped with the continuous extension of $\dpn{\cdot, \cdot}{\phi}$ is a Hilbert space. + \item $(H_\phi, \pi_\phi)$ is a well-defined unitary representation of $G$. + \end{enumerate} + + 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: + \begin{enumerate}[start=2] + \item For each $f, g \in L^1(G; \complex)$, $\pi_\phi(f)(g + N_\phi) = f * g + N_\phi$. + \item There exists a cyclic vector $\xi_\phi \in H_\phi$ for $\pi_\phi$ such that for each $f, g \in L^1(G; \complex)$, + \[ + \dpn{f, g}{\phi} = \dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} + \] + \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 + \[ + \dpn{f, \phi}{L^1(G; \complex)} = \int_G f(x)\Phi(x)dx + \] + + for all $f \in L^1(G; \complex)$. + \end{enumerate} + + Moreover, + \begin{enumerate} + \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 $U\xi_\phi = \eta$. + \end{enumerate} + + + 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 $\phi$}. +\end{definition} +\begin{proof}[Proof, {{\cite[Theorem 3.20]{FollandHarmonic}}}. ] + (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. + + (2): By the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}, for any $f, g \in L^1(G; \complex)$, + \[ + \dpn{f, g}{\phi} \le \dpn{f, f}{\phi}^{1/2} \cdot \dpn{g, g}{\phi}^{1/2} + \] + + In particular, if $g \in N_\phi$, then + \begin{align*} + \dpn{f * g, f * g}{\phi} &= \dpn{g^* * f^* * f * g, \phi}{L^1(G; \complex)} \\ + &= \dpn{g^** (f^**f*g), \phi}{L^1(G; \complex)} = \dpn{f^**f*g, g}{\phi} + \end{align*} + + + so $N_\phi$ is a left ideal of $L^1(G; \complex)$. Let $x \in G$, then $(L_xg)^* = L_{-x}g^*$, and + \begin{align*} + \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)} \\ + &= \dpn{g^* * f, \phi}{L^1(G; \complex)} = \dpn{f, g}{\phi} + \end{align*} + + 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, $\phi \in L^1(G; \complex)^*$ be a positive linear functional, $(H_\phi, \pi_\phi, \xi_\phi)$ be the GNS triple associated with $\phi$. Then: + \begin{enumerate} + \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 + \[ + \dpn{f, \phi}{L^1(G; \complex)} = \int_G f(x)\Phi(x) dx + \] + + 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): 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{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{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 $f, g \in L^1(G; \complex)$, + \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} + \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 $f \in L^1(G; \complex)$, + \[ + \dpn{f, \psi_1}{L^1(G; \complex)} = \lim_{\alpha \in A}\dpn{f, g_\alpha}{\psi_1} = \lim_{\alpha \in A}\lambda\dpn{f, g_\alpha}{\phi} = \lambda \dpn{f, \phi}{L^1(G; \complex)} + \] + + Since $\phi, \psi_1 \in S(G)$, $\lambda = 1$, and $\phi = \psi_1$. As the argument is symmetric, $\phi = \psi_1 = \psi_2$. + + ($\Leftarrow$): 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 + \[ + \dpn{f, \psi_1}{L^1(G; \complex)} = \dpn{\pi_\phi(f)\xi_1, \xi_1}{H_\phi} \quad + \dpn{f, \psi_2}{L^1(G; \complex)} = \dpn{\pi_\phi(f)\xi_2, \xi_2}{H_\phi} + \] + + then $\psi_1, \psi_2 \in S(G)$. Since $P \in \pi_\phi(G)'$, for any $f \in L^1(G; \complex)$, + \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_\phi} + \normn{(1 - P)\xi_\phi}_{H_\phi}^2 \dpn{\pi_\phi(f)\xi_2, \xi_2}{H_\phi} \\ + &= \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)} + \end{align*} + + so $\phi$ is a strict convex combination of two states. If $\psi_1 = \phi$, then for any $f, g \in L^1(G; \complex)$, + \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 + \end{align*} + + so $I - \normn{P\xi_\phi}_{H_\phi}^{-2}P = 0$, which contradicts the fact that $P$ is a non-trivial projection. +\end{proof} \ No newline at end of file diff --git a/src/harmonic/rep/index.tex b/src/harmonic/rep/index.tex index a31315e..0197b2f 100644 --- a/src/harmonic/rep/index.tex +++ b/src/harmonic/rep/index.tex @@ -3,4 +3,6 @@ \input{./unit.tex} \input{./l1.tex} +\input{./gns.tex} +\input{./pos.tex} diff --git a/src/harmonic/rep/pos.tex b/src/harmonic/rep/pos.tex index cb1096b..b36c972 100644 --- a/src/harmonic/rep/pos.tex +++ b/src/harmonic/rep/pos.tex @@ -1,124 +1,6 @@ \section{Positive-Definite Functions} \label{section:positive-definite-function} -\begin{definition}[Positive Linear Functional] -\label{definition:group-l1-positive-functional} - 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 $f \in L^1(G)$. -\end{definition} - - -\begin{definition}[GNS Triple (Locally Compact Group)] -\label{definition:gns-triple-lcg} - Let $G$ be a locally compact group and $\phi \in L^1(G; \complex)^*$. For each $f, g \in L^1(G; \complex)$, let - \[ - \dpn{f, g}{\phi} = \dpn{g^* * f, \phi}{L^1(G; \complex)} - \] - - 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 - \[ - \pi_\phi^0: G \to B(H_\phi^0) \quad \pi_\phi^0(x)(f + N_\phi) = L_xf + N_\phi - \] - - 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: - \begin{enumerate} - \item $H_\phi$ equipped with the continuous extension of $\dpn{\cdot, \cdot}{\phi}$ is a Hilbert space. - \item $(H_\phi, \pi_\phi)$ is a well-defined unitary representation of $G$. - \end{enumerate} - - 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: - \begin{enumerate}[start=2] - \item For each $f, g \in L^1(G; \complex)$, $\pi_\phi(f)(g + N_\phi) = f * g + N_\phi$. - \item There exists a cyclic vector $\xi_\phi \in H_\phi$ for $\pi_\phi$ such that for each $f, g \in L^1(G; \complex)$, - \[ - \dpn{f, g}{\phi} = \dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi} - \] - \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 - \[ - \dpn{f, \phi}{L^1(G; \complex)} = \int_G f(x)\Phi(x)dx - \] - - for all $f \in L^1(G; \complex)$. - \end{enumerate} - - Moreover, - \begin{enumerate} - \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 $U\xi_\phi = \eta$. - \end{enumerate} - - - 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 $\phi$}. -\end{definition} -\begin{proof}[Proof, {{\cite[Theorem 3.20]{FollandHarmonic}}}. ] - (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. - - (2): By the \hyperref[Cauchy-Schwarz inequality]{proposition:cauchy-schwarz}, for any $f, g \in L^1(G; \complex)$, - \[ - \dpn{f, g}{\phi} \le \dpn{f, f}{\phi}^{1/2} \cdot \dpn{g, g}{\phi}^{1/2} - \] - - In particular, if $g \in N_\phi$, then - \begin{align*} - \dpn{f * g, f * g}{\phi} &= \dpn{g^* * f^* * f * g, \phi}{L^1(G; \complex)} \\ - &= \dpn{g^** (f^**f*g), \phi}{L^1(G; \complex)} = \dpn{f^**f*g, g}{\phi} - \end{align*} - - - so $N_\phi$ is a left ideal of $L^1(G; \complex)$. Let $x \in G$, then $(L_xg)^* = L_{-x}g^*$, and - \begin{align*} - \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)} \\ - &= \dpn{g^* * f, \phi}{L^1(G; \complex)} = \dpn{f, g}{\phi} - \end{align*} - - 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}. +\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 + \[ + f * g = \sum_{k = 0}^3 i^k h_k + \] +\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}, + \[ + f * \tilde g = \frac{1}{4}\sum_{k = 0}^3 i^k (f + i^k g) * (f + i^k g)^\sim + \] - 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 + 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 $x, y \in G$, \[ - \dpn{f, \psi_1}{L^1(G; \complex)} = \dpn{\pi_\phi(f)\xi_1, \xi_1}{H_\phi} \quad - \dpn{f, \psi_2}{L^1(G; \complex)} = \dpn{\pi_\phi(f)\xi_2, \xi_2}{H_\phi} + |\phi(x) - \phi(y)|^2 \le 2 - 2\text{Re}\phi(yx^{-1}) \] +\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*} + \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*} - then $\psi_1, \psi_2 \in S(G)$. Since $P \in \pi_\phi(G)'$, for any $f \in L^1(G; \complex)$, + ($\subset$): Let $\phi \in S(G)$ and $V \in \cn_G(1_G)$ be relatively compact, then for each $x \in G$, \begin{align*} - \dpn{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)} + \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*} - - so $\phi$ is a strict convex combination of two states. If $\psi_1 = \phi$, then for any $f, g \in L^1(G; \complex)$, + + 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*} - \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 + |\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. - so $I - \normn{P\xi_\phi}_{H_\phi}^{-2}P = 0$, which contradicts the fact that $P$ is a non-trivial projection. + 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 $g \in C_c(G; \complex)$. + + 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}