The Ext class of an approximately inner automorphism

Abstract

Let A be a simple unital AT algebra of real rank zero. It is shown below that the range of the natural map from the approximately inner automorphism group to KK(A,A) coincides with the kernel of the map KK(A,A)! L 1=0 Hom(Ki(A),Ki(A)). §1 Introduction and preliminaries 1.1: An automorphism of a unital C*-algebra A is said to be an approximately inner automorphism if there is a sequence of unitaries un 2 A such that (a) = limAdun(a) for all a 2 A. It follows that the induced map on K (A) is the identity map; there is, however, an invariant of K-theoretical interest which can occur. Nontrivial extensions may arise in the six-term periodic sequence for the K-theory of the mapping torus. We show below that every extension does arise if A is a simple unital AT algebra of real rank zero. As an immediate corollary we obtain a stronger form of Elliott's classification theorem for such algebras: an invertible KK element that preserves positivity and the class of the unit lifts to an isomorphism. 1.2: Recall that a unital C*-algebra A is said to be a unital AT algebra, if it is expressible as the inductive limit of finite direct sums of algebras of the form, Mn C(T), with unital embeddings. Let A be a unital AT algebra and let be an approximately inner automorphism of A. The mapping torus of is the C*-algebra