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Proceedings of the American Mathematical Society

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Unitary equivalences for essential extensions of $ C^*$-algebras

Author: Huaxin Lin
Journal: Proc. Amer. Math. Soc. 137 (2009), 3407-3420
MSC (2000): Primary 46L05
Published electronically: May 15, 2009
MathSciNet review: 2515410
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Abstract: Let $ A$ be a unital separable $ C^*$-algebra and $ B=C\otimes {\mathcal K},$ where $ C$ is a unital $ C^*$-algebra. Let $ \tau: A\to M(B)/B$ be a unital full essential extension of $ A$ by $ B.$ We show that there is a bijection between elements in a quotient group of $ K_0(B)$ onto the strong unitary equivalence classes of unital full essential extensions $ \sigma$ for which $ [\sigma]=[\tau]$ in $ KK^1(A, B).$ Consequently, when this group is zero, unitarily equivalent full essential extensions are strongly unitarily equivalent. When $ B$ is a non-unital but $ \sigma$-unital simple $ C^*$-algebra with continuous scale, we also study the problem when two approximately unitarily equivalent essential extensions are strongly approximately unitarily equivalent. A group is used to compute the strongly approximate unitary equivalence classes in the same approximate unitary equivalent class of essential extensions.

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Additional Information

Huaxin Lin
Affiliation: Department of Mathematics, East China Normal University, Shanghai, People’s Republic of China
Address at time of publication: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Keywords: Unitary equivalence
Received by editor(s): February 5, 2008
Received by editor(s) in revised form: February 13, 2009
Published electronically: May 15, 2009
Communicated by: Marius Junge
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.