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

Alain J. Valette

Vol. 109 (1983), No. 1, 247–255

DOI: 10.2140/pjm.1983.109.247

Abstract

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

Milestones

Received: 21 January 1982

Revised: 26 June 1982

Published: 1 November 1983