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The real rank zero property of crossed product

Authors:	Xiaochun Fang

Institution:	Department of Applied Mathematics, Tongji University, Shanghai, 200092, People's Republic of China

Abstract:	Let be a unital -algebra, and let $(A, G, \alpha)$ be a -dynamical system with abelian and discrete. In this paper, we introduce the continuous affine map 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 . If $A\times_{\alpha}G$ is of real rank zero and $\hat{G}$ is connected, we have proved that is homeomorphic. Conversely, if 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 is a unital inductive limit of the direct sums of non-elementary simple -algebras of real rank zero, then the $u_{_A}$ above can be cancelled.

Keywords:	Real rank zero crossed product trace state space

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