Journal of Operator Theory

Volume 60, Issue 1, Summer 2008 pp. 113-124.

Topological structure of the unitary group of certain

$C^*$ -algebras

Authors: Bogdan Visinescu
Author institution: Department of Mathematics and Statistics, University of North Florida, 1 UNF Drive Jacksonville, FL 32224, USA and
Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

Summary: Let

$0\rightarrow B\overset{i}{\rightarrow }E\overset{\pi }{\rightarrow }A\rightarrow 0$ be a short exact sequence of

$C^*$ -algebras where

$A$ is a purely infinite simple

$C^*$ -algebra and

$B$ is an essential ideal of

$E$ . In the case

$B$ is the compacts or a nonunital purely infinite simple

$C^*$ -algebra we completely determine the homotopy groups of the unitary group of

$E$ in terms of K-theory. The result can be viewed as a generalization of the well-known Kuiper's theorem to a new class of

$C^*$ -algebras (including certain separable

$C^*$ -algebras).

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