# Journal of Operator Theory

Volume 46, Issue 1, Summer 2001 pp. 99-122.

The Ext class of an approximately inner automorphism. II**Authors**: A. Kishimoto (1), and A. Kumjian

**Author institution:**(1) Department of Mathematics, Hokkaido University, Sapporo 060, Japan

(2) Department of Mathematics, University of Nevada, Reno, NV 89557, USA

**Summary:**Let $A$ be a simple unital AT algebra of real rank zero and Inn($A$) the group of inner automorphisms of $A$. In the previous paper we have shown that the natural map of the group $\Ap$ of approximately inner automorphisms into $\Ext({\rm K}_1(A),{\rm K}_0(A))\oplus\Ext({\rm K}_0(A),{\rm K}_1(A)) $ is surjective; the kernel of this map includes the subgroup of automorphisms which are homotopic to $\Inn$. In this paper we consider the quotient of $\Ap$ by the smaller normal subgroup $\AI$ which consists of asymptotically inner automorphisms and describe it as $\OExt({\rm K}_1(A),{\rm K}_0(A))\oplus \Ext({\rm K}_0(A),{\rm K}_1(A)), $ where $\OExt({\rm K}_1(A),{\rm K}_0(A))$ is a kind of extension group which takes into account the fact that ${\rm K}_0(A)$ is an ordered group and has the usual Ext as a quotient.

**Keywords:**$C^*$-algebra, automorphism, K-theory, extension, trace, asymptotically inner, real rank zero

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