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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
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

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.

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