Crossed product C⁎C⁎-algebras of minimal dynamical systems on the product of the Cantor set and the torus

This paper studies the relationship between minimal dynamical systems on the product of the Cantor set X   and the torus T2T2 and their corresponding crossed product C⁎C⁎-algebras. When the cocycles are rotations, it is shown that the crossed product C⁎C⁎-algebras have tracial rank no more than one, thus these C⁎C⁎-algebras are classifiable by the Elliott invariant. For two such systems, while assuming certain rigidity condition on traces, we prove that if there is certain isomorphism between the ordered K0K0 of the two crossed product C⁎C⁎-algebras, then these two systems are approximately K  -conjugate. Our proof also indicates that C⁎C⁎-strongly flip conjugacy implies approximate K-conjugacy in this case.