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

2014/03/13 by Karen R. Strung, Strung, Karen R.
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Dynamical Systems (math.DS) #FOS: Mathematics #Operator Algebras (math.OA) #Quantum Mechanics and Applications

paper · pdf · doi:10.48550/arxiv.1403.3136

openalex publication_date 2014/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let β: Sⁿ → Sⁿ, for n = 2k + 1, k ≥ 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 (Sⁿ, β) there is a Cantor minimal system (X, α) such that the corresponding product system (X x Sⁿ, αx β) is minimal and the resulting crossed product C*-algebra C(X x Sⁿ) \rtimes_{αx β} 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 Sⁿ) \rtimes_β ℤ induces the same state on the K₀-group and such that the embedding of C(Sⁿ) \rtimes_β ℤ into C(X x Sⁿ) \rtimes_{αx β} ℤ preserves the tracial state space. This implies C(Sⁿ) \rtimes_β 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.

Related

• Minimal dynamical systems on the product of the Cantor set and the circle II
• Crossed Product C^*-algebras of Minimal Dynamical Systems on the Product of the Cantor Set and the Torus
• Minimal dynamics and Z-stable classification
• Classifying crossed product C*-algebras
• Dynamical systems of type (m,n) and their C*-algebras