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 $f \in L^{1}(G; \complex)$
\[\dpn{\pi_\phi(f)\xi_\phi, \xi_\phi}{H_\phi}= \int_{G} f(x)\dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}dx = \int_{G} f(x)\Phi(x)dx\]
by definition of the representation. Since $\pi_{\phi}$ is strong-operator continuous, $\Phi \in C(G; \complex)$.
(2): By Theorem 16.3.3, $\Phi \in BC(G; \complex)$ with $\norm{\Phi}_{u} = \norm{\phi}_{L^1(G; \complex)^*}$. In addition, for any $x \in G$,
\[|\Phi(x)| = |\dpn{\pi_\phi(x)\xi_\phi, \xi_\phi}{H_\phi}| \le \normn{\xi_\phi}_{H_\phi}^{2} = \Phi(1_{G})\]
$\square$
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