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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Diagonal type conditions on group C$^*$-algebras


Authors: Nico Spronk and Peter Wood
Journal: Proc. Amer. Math. Soc. 129 (2001), 609-616
MSC (1991): Primary 22D05, 22D10, 22D25; Secondary 43A65, 43A07, 46L09
Published electronically: July 27, 2000
MathSciNet review: 1800241
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Abstract: Let $G$ be a locally compact group with $\mathbf{C}^* (G)$ and $\mathbf{C}^*_r (G)$ its enveloping and reduced C$^*$-algebras respectively. We show that if $\mathbf{C}^*(G)$ is residually finite dimensional, then $G$is maximally almost periodic, and $\mathbf{C}^*_r (G)$ is residually finite dimensional if and only if $G$ is both amenable and maximally almost periodic. Letting $\lambda_G$ be the left regular representation of $G$, we show that a certain quasidiagonality condition on $\{\lambda_G(s):s\in G\}$ implies that $G$ is amenable.


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Additional Information

Nico Spronk
Affiliation: Department of Pure Mathematics, University of Waterloo, Ontario, Canada N2L 3G1
Email: nspronk@math.uwaterloo.ca

Peter Wood
Affiliation: Department of Pure Mathematics, University of Waterloo, Ontario, Canada N2L 3G1
Email: pwood@math.uwaterloo.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05788-9
PII: S 0002-9939(00)05788-9
Keywords: Group C$^*$-algebra, maximal almost periodicity, residual finite dimensionality, amenability, quasidiagonality
Received by editor(s): April 29, 1999
Published electronically: July 27, 2000
Additional Notes: The first author was partially supported by NSERC
The second author was partially supported by OGS
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society