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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

(APD)-property of $ C^*$-algebras by extensions of $ C^*$-algebras with (APD)


Author: Yifeng Xue
Journal: Proc. Amer. Math. Soc. 135 (2007), 705-711
MSC (2000): Primary 46L05, 46L85, 46L80
Published electronically: October 2, 2006
MathSciNet review: 2262866
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Abstract: A unital $ C^*$-algebra $ \mathcal{A}$ is said to have the (APD)-property if every nonzero element in $ \mathcal{A}$ has the approximate polar decomposition. Let $ \mathcal{J}$ be a closed ideal of $ \mathcal{A}$. Suppose that $ {\tilde{\mathcal{J}}}$ and $ \mathcal{A}\slash\mathcal{J}$ have (APD). In this paper, we give a necessary and sufficient condition that makes $ \mathcal{A}$ have (APD). Furthermore, we show that if $ \mathrm{RR}(\mathcal{J})=0$ and $ \mathrm{tsr}(\mathcal{A}\slash\mathcal{J})=1$ or $ \mathcal{A}\slash\mathcal{J}$ is a simple purely infinite $ C^*$-algebra, then $ \mathcal{A}$ has (APD).


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Additional Information

Yifeng Xue
Affiliation: Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China
Email: yxue3486@hotmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08439-5
PII: S 0002-9939(06)08439-5
Keywords: Approximate polar decomposition, generalized inverse, (WS)--property
Received by editor(s): February 15, 2005
Received by editor(s) in revised form: July 15, 2005
Published electronically: October 2, 2006
Additional Notes: This research was supported by the Natural Science Foundation of China and the Foundation of CSC
Communicated by: David R. Larson
Article copyright: © Copyright 2006 American Mathematical Society