## The real rank zero property of crossed product

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- by Xiaochun Fang
- Proc. Amer. Math. Soc.
**134**(2006), 3015-3024 - DOI: https://doi.org/10.1090/S0002-9939-06-08357-2
- Published electronically: May 8, 2006
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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.## References

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## Bibliographic Information

**Xiaochun Fang**- Affiliation: Department of Applied Mathematics, Tongji University, Shanghai, 200092, People’s Republic of China
- Email: xfang@mail.tongji.edu.cn
- 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
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2231627