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

Runde, Volker

doi:10.1090/S0002-9947-05-03827-4

A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal
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Trans. Amer. Math. Soc. 358 (2006), 391-402 Request permission

Erratum: Trans. Amer. Math. Soc. 367 (2015), 751-754.

Abstract:

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

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