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

 

Remark on certain $ C\sp *$-algebra extensions considered by G. Skandalis


Author: Alexander Kaplan
Journal: Proc. Amer. Math. Soc. 117 (1993), 1047-1050
MSC: Primary 46L05; Secondary 22D25, 46L80
MathSciNet review: 1116262
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Abstract: Let $ \Gamma $ be a nonamenable, discrete ICC subgroup of a connected simple Lie group of real-rank one. G. Skandalis established the exact sequence

$\displaystyle 0 \to K({l^2}(\Gamma )) \to {C^{\ast}}(C_\lambda ^{\ast}(\Gamma ),\;C_\rho ^{\ast}(\Gamma )) \to C_\lambda ^{\ast}(\Gamma \times \Gamma ) \to 0.$

In this note we give sufficient conditions under which such a short exact sequence is not semi-split. In particular, we show that such an extension has no inverse in $ \operatorname{Ext} (C_\lambda ^{\ast}(\Gamma \times \Gamma ))$ provided that the $ {C^{\ast}}$-algebra generated by the unitary representation $ g \to \lambda (g)\rho (g) \otimes \lambda (g)\rho (g)$ of $ \Gamma $ on $ {l^2}(\Gamma ) \otimes {l^2}(\Gamma )$ does not contain nonzero operators from the ideal $ K({l^2}(\Gamma )) \otimes B({l^2}(\Gamma )) + B({l^2}(\Gamma )) \otimes K({l^2}(\Gamma ))$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1116262-8
PII: S 0002-9939(1993)1116262-8
Keywords: $ {C^{\ast}}$-algebra extension, tensor product, ICC-group
Article copyright: © Copyright 1993 American Mathematical Society