## The real rank zero property of crossed product

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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

- P. de la Harpe and G. Skandalis,
*Déterminant associé à une trace sur une algébre de Banach*, Ann. Inst. Fourier (Grenoble)**34**(1984), no. 1, 241–260 (French, with English summary). MR**743629**, DOI 10.5802/aif.958 - Bruce Blackadar,
*$K$-theory for operator algebras*, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR**859867**, DOI 10.1007/978-1-4613-9572-0 - Ola Bratteli, David E. Evans, and Akitaka Kishimoto,
*The Rohlin property for quasi-free automorphisms of the fermion algebra*, Proc. London Math. Soc. (3)**71**(1995), no. 3, 675–694. MR**1347409**, DOI 10.1112/plms/s3-71.3.675 - Lawrence G. Brown and Gert K. Pedersen,
*$C^*$-algebras of real rank zero*, J. Funct. Anal.**99**(1991), no. 1, 131–149. MR**1120918**, DOI 10.1016/0022-1236(91)90056-B - George A. Elliott,
*On the classification of $C^*$-algebras of real rank zero*, J. Reine Angew. Math.**443**(1993), 179–219. MR**1241132**, DOI 10.1515/crll.1993.443.179 - Elliott, G. A., Gong, G. and Li, L.,
*On the Classification of Simple Inductive Limit $C^*$-algebras, II: The Isomorphism Theorem*, preprint. - George A. Elliott and Xiaochun Fang,
*Simple inductive limits of $C^\ast$-algebras with building blocks from spheres of odd dimension*, Operator algebras and operator theory (Shanghai, 1997) Contemp. Math., vol. 228, Amer. Math. Soc., Providence, RI, 1998, pp. 79–86. MR**1667655**, DOI 10.1090/conm/228/03282 - Xiaochun Fang,
*The simplicity and real rank zero property of the inductive limit of continuous trace $C^\ast$-algebras*, Analysis (Munich)**19**(1999), no. 4, 377–389. MR**1743530**, DOI 10.1524/anly.1999.19.4.377 - Akitaka Kishimoto,
*The Rohlin property for automorphisms of UHF algebras*, J. Reine Angew. Math.**465**(1995), 183–196. MR**1344136**, DOI 10.1515/crll.1995.465.183 - Huaxin Lin,
*Almost multiplicative morphisms and some applications*, J. Operator Theory**37**(1997), no. 1, 121–154. MR**1438204** - Huaxin Lin,
*An introduction to the classification of amenable $C^*$-algebras*, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR**1884366**, DOI 10.1142/9789812799883 - Gert K. Pedersen,
*$C^{\ast }$-algebras and their automorphism groups*, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR**548006** - Marc A. Rieffel,
*Dimension and stable rank in the $K$-theory of $C^{\ast }$-algebras*, Proc. London Math. Soc. (3)**46**(1983), no. 2, 301–333. MR**693043**, DOI 10.1112/plms/s3-46.2.301 - Mikael Rørdam,
*Classification of inductive limits of Cuntz algebras*, J. Reine Angew. Math.**440**(1993), 175–200. MR**1225963**, DOI 10.1515/crll.1993.440.175 - Klaus Thomsen,
*Diagonalization in inductive limits of circle algebras*, J. Operator Theory**27**(1992), no. 2, 325–340. MR**1249649** - Klaus Thomsen,
*Traces, unitary characters and crossed products by $\textbf {Z}$*, Publ. Res. Inst. Math. Sci.**31**(1995), no. 6, 1011–1029. MR**1382564**, DOI 10.2977/prims/1195163594 - Jesper Villadsen,
*Simple $C^*$-algebras with perforation*, J. Funct. Anal.**154**(1998), no. 1, 110–116. MR**1616504**, DOI 10.1006/jfan.1997.3168 - Shuang Zhang,
*Matricial structure and homotopy type of simple $C^*$-algebras with real rank zero*, J. Operator Theory**26**(1991), no. 2, 283–312. MR**1225518** - Shuang Zhang,
*On the homotopy type of the unitary group and the Grassmann space of purely infinite simple $C^*$-algebras*, $K$-Theory**24**(2001), no. 3, 203–225. MR**1876798**, DOI 10.1023/A:1013392615189

## Additional 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