(APD)–property of 𝐶*–algebras by extensions of 𝐶*–algebras with (APD)

Xue, Yifeng

doi:10.1090/S0002-9939-06-08439-5

(APD)–property of $C^$–algebras by extensions of $C^$–algebras with (APD)
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Proc. Amer. Math. Soc. 135 (2007), 705-711 Request permission

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).

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