Simple real rank zero algebras with locally Hausdorff spectrum

Ng, Ping

doi:10.1090/S0002-9939-06-07916-0

Simple real rank zero algebras with locally Hausdorff spectrum
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by Ping Wong Ng PDF

Proc. Amer. Math. Soc. 134 (2006), 2223-2228 Request permission

Abstract:

Let $\mathcal {A}$ be a unital, simple, separable $C^*$-algebra with real rank zero, stable rank one, and weakly unperforated ordered $K_0$ group. Suppose, also, that $\mathcal {A}$ can be locally approximated by type I algebras with Hausdorff spectrum and bounded irreducible representations (the bound being dependent on the local approximating algebra). Then $\mathcal {A}$ is tracially approximately finite dimensional (i.e., $\mathcal {A}$ has tracial rank zero). Hence, $\mathcal {A}$ is an $AH$-algebra with bounded dimension growth and is determined by $K$-theoretic invariants. The above result also gives the first proof for the locally $AH$ case.

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