# 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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