Remark on certain $C^ *$-algebra extensions considered by G. Skandalis
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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 \[ 0 \to K({l^2}(\Gamma )) \to {C^{\ast }}(C_\lambda ^{\ast }(\Gamma ),\;C_\rho ^{\ast }(\Gamma )) \to C_\lambda ^{\ast }(\Gamma \times \Gamma ) \to 0.\] 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 $K({l^2}(\Gamma )) \otimes B({l^2}(\Gamma )) + B({l^2}(\Gamma )) \otimes K({l^2}(\Gamma ))$.References
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Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 117 (1993), 1047-1050
- MSC: Primary 46L05; Secondary 22D25, 46L80
- DOI: https://doi.org/10.1090/S0002-9939-1993-1116262-8
- MathSciNet review: 1116262