C⁎C⁎-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere

Let β:Sn→Snβ:Sn→Sn, for n=2k+1n=2k+1, k≥1k≥1, be one of the known examples of a nonuniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (Sn,β)(Sn,β) there is a Cantor minimal system (X,α)(X,α) such that the corresponding product system (X×Sn,α×β)(X×Sn,α×β) is minimal and the resulting crossed product C⁎C⁎-algebra C(X×Sn)⋊α×βZC(X×Sn)⋊α×βZ is tracially approximately an interval algebra (TAI). This entails classification for such C⁎C⁎-algebras. Moreover, the minimal Cantor system (X,α)(X,α) is such that each tracial state on C(X×Sn)⋊α×βZC(X×Sn)⋊α×βZ induces the same state on the K0K0-group and such that the embedding of C(Sn)⋊βZC(Sn)⋊βZ into C(X×Sn)⋊α×βZC(X×Sn)⋊α×βZ preserves the tracial state space. This implies C(Sn)⋊βZC(Sn)⋊βZ is TAI after tensoring with the universal UHF algebra, which in turn shows that the C⁎C⁎-algebras of these examples of minimal diffeomorphisms of odd dimensional spheres are classified by their tracial state spaces.