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I am reading the paper [Margulis, G. A.; Nevo, A.; Stein, E. M., Analogs of Wiener's ergodic theorems for semisimple Lie groups. II. Duke Math. J. 103 (2000), no. 2, 233–259] (MSN).

I want to understand the following argument. Let $(G,K)$ be a Gelfand pair. Then $M(G,K)$ i.e. the set of all bi-$K$-invariant measures on $G$ is a commutative Banach algebra. Suppose $\tau:G\to B(\mathcal H_\tau)$ be a strongly continuous unitary representation of $G.$ Denote $A_{\tau}:=\overline{\tau(M(G,K)}.$ Then spectrum of $A_\tau$ can be identified with all positive definite spherical functions on $G.$ How do you see that?

Secondly given a probability measure $\mu\in M(G,K)$ define $$\|\mu\|_T:=\sup\{|\phi_\lambda(\mu)|:\text{$\phi_\lambda$ positive-definite spherical function}\}.$$ Then $\|\mu\|_T\geq \tau(\mu).$

I do not understand the identification of spectrum of $A_\tau$ with positive-definite spherical functions. Secondly, what does one mean by $\phi_\lambda(\mu)$ since $\phi_\lambda$ is supposed to be a function on $G$?

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    $\begingroup$ Did you check the four references given for the identification? $\endgroup$
    – LSpice
    Commented Apr 21, 2020 at 17:58
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    $\begingroup$ Also, $\phi_\lambda(\mu)$ means that $\phi_\lambda$ is viewed as a character of $\overline{\tau(M(G, K))}$, then evaluated at $\tau(\mu)$. $\endgroup$
    – LSpice
    Commented Apr 21, 2020 at 18:07
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    $\begingroup$ Anyway, (1) notice that the spectrum is not identified with all pdsf's, but only a subset, and (2) I think that the identification is the simple one: given a character $\chi$ of $\overline{\tau(M(G, K))}$, send $g \in G$ to the value of $\chi \circ \tau$ at the unit mass on $K g K$. $\endgroup$
    – LSpice
    Commented Apr 21, 2020 at 18:17
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    $\begingroup$ The closed span of the various unit $K g K$'s is $M(G, K)$, so injectivity follows. (Also, you can delete comments.) $\endgroup$
    – LSpice
    Commented Apr 21, 2020 at 18:41
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    $\begingroup$ $G$ is a (finite-dimensional) Lie group, so it is $\sigma$-compact. $\endgroup$
    – LSpice
    Commented Apr 21, 2020 at 19:04

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