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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Volker Runde
Journal: Trans. Amer. Math. Soc. 358 (2006), 391-402
MSC (2000): Primary 46H20; Secondary 22A15, 22A20, 43A07, 43A10, 43A60, 46H25, 46M18, 46M20
Published electronically: July 26, 2005
MathSciNet review: 2171239
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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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Additional Information

Volker Runde
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: vrunde@ualberta.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03827-4
PII: S 0002-9947(05)03827-4
Keywords: Locally compact groups, Connes-amenability, normal, virtual diagonals, weakly almost periodic functions, semigroup compactifications, minimally weakly almost periodic groups
Received by editor(s): October 26, 2003
Received by editor(s) in revised form: June 1, 2004
Published electronically: July 26, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.