Crossed Product $C^*$-algebras of Minimal Dynamical Systems on the Product of the Cantor Set and the Torus

Abstract: This paper studies the relationship between minimal dynamical systems on the product of the Cantor set ($X$) and torus ($\T^2$) and their corresponding crossed product $C^*$-algebras. For the case when the cocycles are rotations, we studied the structure of the crossed product $C^*$-algebra $A$ by looking at a large subalgebra $A_x$. It is proved that, as long as the cocycles are rotations, the tracial rank of the crossed product $C^*$-algebra is always no more than one, which then indicates that it falls into the category of classifiable $C^*$-algebras. If a certain rigidity condition is satisfied, it is shown that the crossed product $C^*$-algebra has tracial rank zero. Under this assumption, it is proved that for two such dynamical systems, if $A$ and $B$ are the corresponding crossed product $C^*$-algebras, and we have an isomorphism between $K_i(A)$ and $K_i(B)$ which maps $K_i(C(X \times \T^2))$ to $K_i(C(X \times \T^2))$, then these two dynamical systems are approximately $K$-conjugate. The proof also indicates that $C^*$-strongly flip conjugacy implies approximate $K$-conjugacy in this case.