Let $ϕ_f$ be a Furstenberg transformation on the 2-torus $𝕋^2$ defined by $ϕ_{f}(x,y) = (e^{2πiθ}x, e^{2πif(x)}xy) for any x,y ∈ 𝕋, where θ is an irrational number and f is a real-valued continuous function on the 1-torus 𝕋. Let $A(ϕ_{f})$ be the crossed product associated with $ϕ_{f}$. We show that $A(ϕ_{f})$ has a unique tracial state for any irrational number θ and any real-valued continuous function f on 𝕋.
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Following Jansen and Waldmann, and Kajiwara and Watatani, we introduce notions of coactions of a finite-dimensional C*-Hopf algebra on a Hilbert C*-bimodule of finite type in the sense of Kajiwara and Watatani and define their crossed product. We investigate their basic properties and show that the strong Morita equivalence for coactions preserves the Rokhlin property for coactions of a finite-dimensional C*-Hopf algebra on unital C*-algebras.
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