The real rank zero property of crossed product

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.