(APD)–property of $C^*$–algebras by extensions of $C^*$–algebras with (APD)
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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 $\mathcal {A}/\mathcal {J}$ 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 $\mathrm {tsr}(\mathcal {A}/\mathcal {J})=1$ or $\mathcal {A}/\mathcal {J}$ is a simple purely infinite $C^*$–algebra, then $\mathcal {A}$ has (APD).References
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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
- 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
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 705-711
- MSC (2000): Primary 46L05, 46L85, 46L80
- DOI: https://doi.org/10.1090/S0002-9939-06-08439-5
- MathSciNet review: 2262866