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The real rank zero property of crossed product


Author: Xiaochun Fang
Journal: Proc. Amer. Math. Soc. 134 (2006), 3015-3024
MSC (2000): Primary 46L05; Secondary 46L35, 46L40
DOI: https://doi.org/10.1090/S0002-9939-06-08357-2
Published electronically: May 8, 2006
MathSciNet review: 2231627
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Abstract: Let $A$ be a unital $C^*$-algebra, and let $(A, G, \alpha )$ be a $C^*$-dynamical system with $G$ abelian and discrete. In this paper, we introduce the continuous affine map $R$ from the trace state space $T(A\times _{\alpha }G)$ of the crossed product $A\times _{\alpha }G$ to the $\alpha$-invariant trace state space $T(A)_{\alpha ^*}$ of $A$. If $A\times _{\alpha }G$ is of real rank zero and $\hat {G}$ is connected, we have proved that $R$ is homeomorphic. Conversely, if $R$ is homeomorphic, we also get some properties and real rank zero characterization of $A\times _{\alpha }G$. In particular, in that case, $A\times _{\alpha }G$ is of real rank zero if and only if each unitary element in $A\times _{\alpha }G$ with the form $u_{_A}\prod _{i=1}^n x_i^*y_i^*x_iy_i$ can be approximated by the unitary elements in $A\times _{\alpha }G$ with finite spectrum, where $u_{_A}\in U_0(A)$, $x_i,y_i\in C_c(G,A)\cap U_0(A\times _{\alpha }G)$, and if moreover $A$ is a unital inductive limit of the direct sums of non-elementary simple $C^*$-algebras of real rank zero, then the $u_{_A}$ above can be cancelled.


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Additional Information

Xiaochun Fang
Affiliation: Department of Applied Mathematics, Tongji University, Shanghai, 200092, People’s Republic of China
Email: xfang@mail.tongji.edu.cn

Keywords: Real rank zero, crossed product, trace state space
Received by editor(s): January 3, 2005
Received by editor(s) in revised form: May 9, 2005
Published electronically: May 8, 2006
Additional Notes: This article was supported by the National Natural Science Foundation of China (10271090).
Communicated by: David R. Larson
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.