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Property T and Amenable Transformation Group C*-algebras

Published online by Cambridge University Press:  20 November 2018

F. Kamalov*
Affiliation:
Mathematics Department, Canadian University of Dubai, Dubai, UAE. e-mail: firuz@cud.ac.ae
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Abstract

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It is well known that a discrete group that is both amenable and has Kazhdan’s Property $T$ must be finite. In this note we generalize this statement to the case of transformation groups. We show that if $G$ is a discrete amenable group acting on a compact Hausdorff space $X$, then the transformation group ${{C}^{*}}$-algebra ${{C}^{*}}\left( X,\,G \right)$ has Property $T$ if and only if both $X$ and $G$ are finite. Our approach does not rely on the use of tracial states on ${{C}^{*}}\left( X,\,G \right)$.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Anantharaman-Delaroche, C., Cohomology of property T groupoids and applications. Ergodic Theory Dynam. Systems 25 (2005), no. 4, 9771013. http://dx.doi.org/10.1017/S0143385704000884 Google Scholar
[2] Bekka, B., Property (T) for C*-algebras. Bull. London Math. Soc. 38 (2006), no. 5, 857867. http://dx.doi.org/10.1112/S0024609306018765 CrossRefGoogle Scholar
[3] Brown, N. P., Kazhdan's property T and C*-algebras. J. Funct. Anal. 240 (2006), no. 1, 290296. http://dx.doi.org/10.1016/j.jfa.2006.05.003 CrossRefGoogle Scholar
[4] Kazhdan, D. A., Connection of the dual space of a group with the structure of its closed subgroups. Funct. Anal. Appl. 1 (1967), 6365. CrossRefGoogle Scholar
[5] Leung, C.-W. and Ng, C.-K., Property (T) and strong property (T) for unital C*-algebras. J. Funct. Anal. 256 (2009), no. 9, 30553070. http://dx.doi.org/10.1016/j.jfa.2009.01.004 CrossRefGoogle Scholar
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