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

Let β: S^n \to S^n, for n = 2k + 1, k \geq 1, be one of the known examples of a non-uniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (S^n, β) there is a Cantor minimal system (X, α) such that the corresponding product system (X x S^n, αx β) is minimal and the resulting crossed product C*-algebra C(X x S^n) \rtimes_{αx β} \mathbb{Z} is tracially approximately an interval algebra (TAI). This entails classification for such C*-algebras. Moreover, the minimal Cantor system (X, α) is such that each tracial state on C(X x S^n) \rtimes_β \mathbb{Z} induces the same state on the K_0-group and such that the embedding of C(S^n) \rtimes_β \mathbb{Z} into C(X x S^n) \rtimes_{αx β} \mathbb{Z} preserves the tracial state space. This implies C(S^n) \rtimes_β \mathbb{Z} is TAI after tensoring with the universal UHF algebra, which in turn shows that the C*-algebras of these examples of minimal diffeomorphisms of odd dimensional spheres are classified by their tracial state spaces.