Crossed Products Of C~*-algebras

In this paper,we introduce certain tracial Rokhlin property for automorphisms on non-simple C^*-algebras which is similar to the one for automorphisms on simple C^*-algebras.We prove:let A be a unital AF-algebra and letαbe an automorphism on A.Suppose that a has the tracial Rokhlin property and A isα-simple.Suppose also there is an integer J≥1 such thatα_*0^J =id_K0(A),then A(?)_αZ has tracial rank zero.In the second section,we prove that for certain C^*-algebra class A which is lager than the C^*-algebra class of tracial rank zero(here elements of A need not have real rank zero),we can get the following conclusion:let A be a unital simple amenable C^*-algebra in A and letαbe an automorphism on A.Suppose thatαhas Rokhlin property and there is an integer J≥1 such thatα_*0^J｜G =id_G,where G is a subgroup of K₀(A) such thatρ_A(G) is dense inρ_A(K₀(A)),then A(?)_αZ∈A.In the third section,we prove the following rezult:let X be a Cantor set,let A be a unital separable simple C^*-algebra with tracial rank zero which satisfies the UCT,and letαbe an automorphism on C(X,A).Suppose C(X,A) isα-simple and satisfies some K-theoretical conditions,then C(X,A)(?)_αZ has tracial rank zero.In the forth section,we prove the following rezult:let X be an infinite compact metric space with finite covering dimension,let A be a unital simple AF-algebra,and letαbe an automorphism on C(X,A).Suppose C(X,A) isα-simple and satisfies some K-theoretical conditions,then C(X,A)(?)_αZ has tracial rank zero.