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A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal

Volker Runde
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Abstract

Let $G$ be a locally compact group, and let $WAP(G)$ denote the space of weakly almost periodic functions on $G$. We show that, if $G$ is a $[SIN]$-group, but not compact, then the dual Banach algebra $WAP(G)^\ast$ does not have a normal, virtual diagonal. Consequently, whenever $G$ is an amenable, non-compact $[SIN]$-group, $WAP(G)^\ast$ is an example of a Connes-amenable, dual Banach algebra without a normal,virtual diagonal.

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Volker Runde. 2004-06-01. A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal. https://arxiv.org/abs/math/0310151

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