Classification of homomorphisms and dynamical systems

By Huaxin Lin. 2004

53 citations recorded by the source corpus.

Let A be a unital simple C*-algebra, with tracial rank zero and let X be a compact metric space. Suppose that h 1 ,h 2 : C(X) → A are two unital monomorphisms. We show that h 1 and h 2 are approximately unitarily equivalent if and only if [h 1 ] = [h 2 ] in KL(C(X), A) and τ o h 1 (f) = τ o h 2 (f) for every f ∈ C(X) and every trace r of A. Inspired by a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let X be a compact metric space and let 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 the 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 a ∈ Aut(A). We prove that if α r fixes a large subgroup of K 0 (A) and has the tracial Rokhlin property, then Ax α Z is again a unital simple AH-algebra with no dimension growth and with real rank zero.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8ce6298cc0c66f02af4b23257b8bfa6257de4f92

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

A classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one. (2018, unknown): “6 of [70] and 2.”
Approximate Homotopy of Homomorphisms from C(x) Into a Simple C*-Algebra (2006, background): “6 of [32] associated with ǫ/4, G′ 0 and σ above.”