Full extensions and approximate unitary equivalence

Huaxin Lin

Vol. 229 (2007), No. 2, 389–428

DOI: 10.2140/pjm.2007.229.389

Abstract

Let A be a unital separable amenable C^∗-algebra and let C be a unital C^∗-algebra with a certain infinite property. We show that two full monomorphisms h₁,h₂ : A → C are approximately unitarily equivalent if and only if [h₁] = [h₂] in KL(A,C). Let B be a nonunital but σ-unital C^∗-algebra for which M(B)∕B has a certain infinite property. We prove that two full essential extensions are approximately unitarily equivalent if and only if they induce the same element in KL(A,M(B)∕B). The set of approximately unitarily equivalence classes of full essential extensions forms a group. If A satisfies the Universal Coefficient Theorem, the group can be identified with KL(A,M(B)∕B).

Keywords

extension of C^∗-algebras, simple C^∗-algebras

Mathematical Subject Classification 2000

Primary: 46L05, 46L35

Milestones

Received: 16 April 2004

Accepted: 12 December 2005

Published: 1 February 2007