Minimal dynamical systems on the product of the Cantor set and the circle

Abstract

We prove that a crossed product algebra arising from a minimal dynamical system on the product of the Cantor set and the circle has real rank zero if and only if that system is rigid. In the case that cocycles take values in the rotation group, it is also shown that rigidity implies tracial rank zero, and in particular, the crossed product algebra is isomorphic to a unital simple AT-algebra of real rank zero. Under the same assumption, we show that two systems are approximately $K$-conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps $C(X\times \T)$ onto $C(X\times \T)$.

44 pages

References in corpus (8)

• Crossed products by finite cyclic group actions with the tracial Rokhlin property
• Minimal dynamical systems on the product of the Cantor set and the circle
• Crossed products of the Cantor set by free minimal actions of Z^d
• Direct limit decomposition for C*-algebras of minimal diffeomorphisms
• Cancellation and stable rank for direct limits of recursive subhomogeneous algebras
• Classification of simple C*-algebras and higher dimensional noncommutative tori
• Furstenberg transformations on irrational rotation algebras
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Cited by in corpus (12)

• Minimal dynamical systems on the product of the Cantor set and the circle
• The Automorphism group of a simple tracially AI algebra
• Minimal dynamical systems on connected odd dimensional spaces
• Approximate Unitary Equivalence in Simple C^*-algebras of Tracial Rank One
• Crossed Product $C^{\ast }$-algebras of Minimal Dynamical Systems on the Product of the Cantor Set and the Torus
• Minimal Homeomorphisms and Approximate Conjugacy in Measure
• Crossed products by minimal homeomorphisms
• Approximate conjugacy and full groups of Cantor minimal systems
• C*-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere
• The structure of crossed products by automorphisms of $C(X,D)$
• Furstenberg Transformations and Approximate Conjugacy
• Classification of homomorphisms and dynamical systems