32.3 The GNS Construction for Groups
Definition 32.3.1 (Positive Linear Functional).label Let $G$ be a locally compact group and $\phi \in L^{1}(G; \complex)^{*}$[1], then $\phi$ is positive if $\dpn{f^* * f, \phi}{L^1(G; \complex)}\ge 0$ for all $f \in L^{1}(G)$.
Definition 32.3.2 (State).label Let $G$ be a locally compact group and $\phi \in L^{1}(G; \complex)^{*}$ be a positive linear functional, then $\phi$ is a state if $\norm{\phi}_{L^1(G; \complex)^*}= 1$. The set $S(G)$ of all states on $L^{1}(G; \complex)$ is the state space of $G$.
Definition 32.3.3 (GNS Triple (Locally Compact Group)).label 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
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
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:
- (1)
$H_{\phi}$ equipped with the continuous extension of $\dpn{\cdot, \cdot}{\phi}$ is a Hilbert space.
- (2)
$(H_{\phi}, \pi_{\phi})$ is a well-defined unitary representation of $G$.
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:
- (3)
For each $f, g \in L^{1}(G; \complex)$, $\pi_{\phi}(f)(g + N_{\phi}) = f * g + N_{\phi}$.
- (4)
There exists a cyclic vector $\xi_{\phi} \in H_{\phi}$ for $\pi_{\phi}$ such that for each $f, g \in L^{1}(G; \complex)$,
\[\dpn{f, g}{\phi}= \dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi}\] - (5)
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 $f \in L^{1}(G; \complex)$.
Moreover,
- (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 $U\xi_{\phi} = \eta$.
The representation $(H_{\phi}, \pi_{\phi})$ is the unitary cyclic representation of $G$ induced by $\phi$, and the triple $(H_{\phi}, \pi_{\phi}, \xi_{\phi})$ is the Gelfand-Naimark-Segal (GNS) triple associated with $\phi$.
Proof, [Theorem 3.20, Fol16]. (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 Cauchy-Schwarz inequality, for any $f, g \in L^{1}(G; \complex)$,
In particular, if $g \in N_{\phi}$, then
so $N_{\phi}$ is a left ideal of $L^{1}(G; \complex)$. Let $x \in G$, then $(L_{x}g)^{*} = L_{-x}g^{*}$, and
by Proposition 31.4.9. 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 continuity of translation.
(3): Let $f, g, h \in L^{1}(G; \complex)$, then after passing through Riemann sums,
As the above holds for all $h \in L^{1}(G; \complex)$, $\pi_{\phi}(f)(g + N_{\phi}) = f * g + N_{\phi}$.
(4): Using Proposition 31.4.10, 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
then $\normn{\xi_\phi}_{H_\phi}\le \norm{\phi}_{L^1(G; \complex)^*}^{1/2}$. Moreover, for any $f, g \in L^{1}(G; \complex)$,
$\square$
Corollary 32.3.4.label 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:
- (1)
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 $f \in L^{1}(G; \complex)$.
- (2)
$\Phi(1_{G}) = \norm{\Phi}_{u} = \norm{\phi}_{L^1(G; \complex)^*}$.
Proof. (1): For any $f \in L^{1}(G; \complex)$
by definition of the representation. Since $\pi_{\phi}$ is strong-operator continuous, $\Phi \in C(G; \complex)$.
(2): By Theorem 16.3.3, $\Phi \in BC(G; \complex)$ with $\norm{\Phi}_{u} = \norm{\phi}_{L^1(G; \complex)^*}$. In addition, for any $x \in G$,
$\square$
Proposition 32.3.5.label 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 $S(G)$.
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
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 $f, g \in L^{1}(G; \complex)$,
so $T \in \pi_{\phi}(G)'$. By Schur’s Lemma, there exists $\lambda \in \complex \setminus \bracsn{0}$ such that $T = \lambda I$.
Using Proposition 31.4.10, let $\angles{g_\alpha}_{\alpha \in A}\subset L^{1}(G; \complex)$ be an approximate identity for $L^{1}(G; \complex)$, then for any $f \in L^{1}(G; \complex)$,
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 Schur’s Lemma, $\pi_{\phi}(G)'$ is non-trivial. Since $\pi_{\phi}(G)'$ is a von Neumann algebra, it admits a non-trivial projection $P \in B(H_{\phi})$ by Theorem 39.6.2.
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
then $\psi_{1}, \psi_{2} \in S(G)$. Since $P \in \pi_{\phi}(G)'$, for any $f \in L^{1}(G; \complex)$,
so $\phi$ is a strict convex combination of two states. If $\psi_{1} = \phi$, then for any $f, g \in L^{1}(G; \complex)$,
so $I - \normn{P\xi_\phi}_{H_\phi}^{-2}P = 0$, which contradicts the fact that $P$ is a non-trivial projection.$\square$
- Due to technical difficulties, positive linear functionals are a priori not represented by $L^{\infty}$ functions. keyboard_return
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