The weak closure of the set of left translation operators

Chou, Ching; Xu, Guangwu

doi:10.1090/S0002-9939-99-04537-2

The weak closure of the set of left translation operators
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by Ching Chou and Guangwu Xu PDF

Proc. Amer. Math. Soc. 127 (1999), 465-471 Request permission

Abstract:

It is known that for an amenable locally compact group $G$, $0$ is not in the weak closure of $\{ \lambda (g) : g \in G \}$ of $VN(G)$. In this paper, it is proved that the converse of this is true. In other words, if $G$ is a non-amenable locally compact group, then $0$ is in the weak closure of $\{ \lambda (g) : g \in G \}$. This answers several questions of Ülger. Applications to the algebra $C^{*}_{\delta }(G)$ and the dual of the reduced group $C^{*}$-algebra are obtained.

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