2004/04/01 by Huaxin Lin, Lin, Huaxin
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA
paper · pdf · doi:10.48550/arxiv.math/0404018
openalex publication_date 2004/04/01 · arxiv created 2004/11/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A be a unital simple C*-algebra with tracial rank zero and X be a compact metric space. Suppose that h1, h2: C(X)→ A are two unital monomorphisms. We show that h1 and h2 are approximately unitarily equivalent if and only if [h1]=[h2] \rm in KL(C(X),A) \rm and τ∘ h1(f)=τ∘ h2(f) for every f∈ C(X) and every trace τ of A. Adopting a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let X be a compact metric space and α, β: X→ X be two minimal homeomorphisms. Using the above mentioned result, we show that two dynamical systems are approximately conjugate in that sense if and only if a K-theoretical condition is satisfied. In the case that X is the Cantor set, this notion coincides with strong orbit equivalence of Giordano, Putnam and Skau and the K-theoretical condition is equivalent to saying that the associate crossed product C*-algebras are isomorphic. Another application of the above mentioned result is given for C^*-dynamical systems related to a problem of Kishimoto. Let A be a unital simple AH-algebra with no dimension growth and with real rank zero, and let α∈ Aut(A). We prove that if αr fixes a large subgroup of K0(A) and has the tracial Rokhlin property then A\rtimesα\Z is again a unital simple AH-algebra with no dimension growth and with real rank zero.