Vol. 109, No. 1, 1983

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A remark on the Kasparov groups Exti(A, B)

Alain J. Valette

Vol. 109 (1983), No. 1, 247–255

Let A and B be C-algebras. We show that, under reasonable assumptions (A unital, nuclear and separable, B with a strictly positive element), the groups Exti(A,B) of Kasparov are isomorphic—up to a shift of dimension—to the K-theory groups of some commutant of A in the outer multiplier algebra of B ⊗𝒦. The main tool to prove this is Kasparov’s “generalized theorem of Voiculescu”. Following an idea of Paschke, we use our result to get a part of the “generalized Pimsner-Voiculescu exact sequence” for crossed products.

Mathematical Subject Classification 2000
Primary: 46L80
Secondary: 46L05, 46M20
Received: 21 January 1982
Revised: 26 June 1982
Published: 1 November 1983
Alain J. Valette
Institut de mathématiques, Faculté des Sciences
Université de Neuchâtel
Rue Emile-Argand 11
CH Neuchâtel