Journal of Operator Theory

Volume 61, Issue 1, Winter 2009 pp. 147-169.

Simple unital *-algebras with the stable locally finite dimensional property

Authors: P.W. Ng (1), Z. Niu (2), and E. Ruiz (3)
Author institution: (1) Dept. of Mathematics, University of Louisiana, 217 Maxim D. Doucet Hall, P. O. Box 41010, Lafayette, LA, 70504-1010, USA
(2) Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada
(3) Dept. of Mathematics, University of Toronto, 40 St. George St., Toronto, Ontario, M5S 2E4, Canada

Summary: The stable locally finite dimensional (SLF) property is the ``opposite'' of the tracial rank zero (TR0) property (defined by Lin). We conjecture that for unital separable simple $C^*$-algebras, SLF is equivalent to nuclearity and TR0. We prove the following: \begin{vartheorem}[Theorem] Let $\A$ be a unital simple separable $C^*$-algebra. \nr{i} If $\A$ is nuclear and has {\rm TR0} then $\A$ is {\rm SLF}. \nr{ii} If $\A$ is {\rm SLF} then $\A$ is nuclear, quasidiagonal, and has real rank zero, stable rank one and weakly unperforated $K_0$ group. \end{vartheorem} We also show that if $\A$ is a unital simple separable $C^*$-algebra with the SLF property, then every embedding of a commutative $C^*$-algebra into $\A$ has a {\rm TR0} type property.

Keywords: Stable locally finite dimensional, quasidiagonal, nuclear, tracial rank zero, $K$-theory

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