Revisions to Kazhdan Property T of semisimple Lie groups

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edited Apr 21, 2020 at 17:58

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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.Analogs of Wiener's ergodic theorems for semisimple Lie groups. II. Duke Math. J. 103 (2000), no. 2, 233–259.]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 $G$?

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.].

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 $G$?

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 $G$?

resumed edits erased by collision of edits...

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edited Apr 21, 2020 at 17:01

YCor

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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.].

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}\}$.$$\|\mu\|_T:=\sup\{|\phi_\lambda(\mu)|:\text{$\phi_\lambda$ positive-definite spherical function}\}.$$ 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 $G$!!??

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.]. 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 $G$!!??

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.].

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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edited Apr 21, 2020 at 16:26

A beginner mathmatician

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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.]. 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 afunctiona function on $G$!!??

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.]. 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 afunction on $G$!!??

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.]. 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 $G$!!??