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Property T for general C*-algebras

Published online by Cambridge University Press:  29 November 2013

CHI–KEUNG NG
CHI–KEUNG NG*
Affiliation:
Chern Institute of Mathematics, Nankai University, Tianjin 300071, China. e-mail: ckng@nankai.edu.cn
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Abstract

In this paper, we extend the definition of property T and strong property T to general C*-algebras (not necessarily unital). We show that if an inclusion pair of locally compact groups (G,H) has property T, then (C*(G), C*(H)) has property T. As a partial converse, if T is abelian and C*(G) has property T, then T is compact. We also show that if Ω is a first countable locally compact Hausdorff space, then C0(Ω) has (strong) property T if and only if Ω is discrete. Furthermore, the non-unital C*-algebra $c_0(\mathbb{Z}^n)\rtimes SL_n(\mathbb{Z})$ has strong property T when n ≥ 3. We also give some equivalent forms of strong property T, which are new even in the unital case.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 156 , Issue 2 , March 2014 , pp. 229 - 239
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Aarne, J. F.On the Mackey-topology for a von Neumann algebra. Math. Scand. 22 (1968), 87–107.Google Scholar
[2]Bekka, M. B.Property (T) for C*-algebras. Bull. London Math. Soc. 38 (2006) 857–867.CrossRefGoogle Scholar
[3]Bekka, M. B., Harpe, P. and Valette, A.Kazhdan's Property (T), (Cambridge University Press, 2008).Google Scholar
[4]Brown, N. P.Kazhdan's property (T) and C*-algebras. J. Funct. Anal. 240 (2006) 290–296.CrossRefGoogle Scholar
[5]Connes, A. and Jones, V.Property T for von Neumann algebras. Bull. Lond. Math. Soc. 17 (1985), 57–62.CrossRefGoogle Scholar
[6]Jolissaint, P.On property (T) for Pairs of topological groups, Enseign. Math. (2) 51 (2005), 31–45.Google Scholar
[7]Kazhdan, D.Connection of the dual space of a group with the structure of its closed subgroups. Funct. Anal. Appl. 1 (1967), 63–65.Google Scholar
[8]Leung, C. W. and Ng, C. K.Property (T) and strong Property (T) for unital C*-algebras. J. Func. Anal. 256 (2009), 3055–3070.CrossRefGoogle Scholar
[9]Leung, C. W., Ng, C. K. and Wong, N. C.Property (T) for non-unital C*-algebras. J. Math. Anal. Appl. 341 (2008), 1102–1106.Google Scholar
[10]Li, D.Property T** for C*-algebras. Acta Math. Sinica. 28 (2012), 1845–1850.Google Scholar
[11]Schaefer, H. H.Topological Vector spaces. GTM 3 (Springer-Verlag, 1971).Google Scholar
[12]Sentilles, F. D. and Taylor, D. C.Factorization in Banach algebras and the general strict topology. Trans. Amer. Math. Soc. 142 (1969), 141–152.Google Scholar
[13]Takesaki, M.Theory of Operator Algebras I (Springer-Verlag, New York, 1979).CrossRefGoogle Scholar
[14]Taylor, D. C.The strict topology for double centralizer algebras. Trans. Amer. Math. Soc. 150 (1970), 633–643.Google Scholar