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

 

On the invariant translation approximation property for discrete groups


Author: Joachim Zacharias
Journal: Proc. Amer. Math. Soc. 134 (2006), 1909-1916
MSC (2000): Primary 46L06, 46L85, 20F69
Published electronically: January 31, 2006
MathSciNet review: 2215118
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Abstract: Recently J.Roe considered the question of whether for a discrete group the reduced group $ C^*$-algebra $ C_r^*(\Gamma)$ is the fixed point algebra of $ \{$Ad$ (\rho_t) \mid t \in \Gamma \}$ acting on the uniform Roe algebra $ UC_r^*(\Gamma)$. $ \Gamma$ is said to have the invariant translation approximation property in this case. We consider a slight generalization of this property which, for exact $ \Gamma$, is equivalent to the operator space approximation property of $ C_r^*(\Gamma)$. We also give a new characterization of exactness and a short proof of the equivalence of exactness of $ \Gamma$ and exactness of $ C_r^*(\Gamma)$ for discrete groups.


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Additional Information

Joachim Zacharias
Affiliation: School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
Email: jz@maths.nott.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08191-3
PII: S 0002-9939(06)08191-3
Keywords: Exact groups, uniform Roe algebra, invariant translation approximation property, operator approximation property
Received by editor(s): March 22, 2004
Received by editor(s) in revised form: February 1, 2005
Published electronically: January 31, 2006
Communicated by: David R. Larson
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.