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. (1)

   $H_{\phi}$ equipped with the continuous extension of $\dpn{\cdot, \cdot}{\phi}$ is a Hilbert space.

2. (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:

1. (3)

   For each $f, g \in L^{1}(G; \complex)$, $\pi_{\phi}(f)(g + N_{\phi}) = f * g + N_{\phi}$.

2. (4)

   There exists a cyclic vector $\xi_{\phi} \in H_{\phi}$ for $\pi_{\phi}$ such that for each $f, g \in L^{1}(G; \complex)$,

   \[\dpn{f, g}{\phi}= \dpn{\pi_\phi(f)\xi_\phi, \pi_\phi(g)\xi_\phi}{H_\phi}\]

   
   
3. (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,

1. (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 $\phi$.