Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The real rank zero property of crossed product
HTML articles powered by AMS MathViewer

by Xiaochun Fang PDF
Proc. Amer. Math. Soc. 134 (2006), 3015-3024 Request permission

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
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46L05, 46L35, 46L40
  • Retrieve articles in all journals with MSC (2000): 46L05, 46L35, 46L40
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