The real rank zero property of crossed product

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 a. ne map R from the trace state space T( A x (alpha)G) of the crossed product A x (alpha)G to the alpha-invariant trace state space T(A)(alpha)* of A. If A x (alpha)G is of real rank zero and (G) over cap 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 x (alpha)G. In particular, in that case, A x (alpha)G is of real rank zero if and only if each unitary element in A x (alpha)G with the form u(A) Pi(n)(i=1) x(i)(*) y(i)(*) x(i)y(i) can be approximated by the unitary elements in A x (alpha)G with finite spectrum, where u(A)is an element of U-0(A), x(i), y(i) is an element of C-c(G, A) boolean AND U-0(A x (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.