On dualities of actions

2022/08/22 by Hyun Ho Lee, Lee, Hyun Ho
Mathematics · Physics and Astronomy · #46L35 #46L55(Primary) #46L80(Secondary) #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2208.10058

openalex publication_date 2022/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce the notion of the weak tracial approximate representability of a discrete group action on a unital C^*-algebra which could have no projections like the Jiang-Su algebra Z. Then we show a duality between the weak tracial Rokhlin property and the weak tracial approximate representability. More precisely, when G is a finite abelian group and α:G\curvearrowright A is a group action on a unital simple infinite dimensional C^*-algebra, we prove that 1. α has the weak tracial Rokhlin property if and only if α has the weak tracial approximate representability. 2. α has the weak tracial approximate representability if and only if α has the weak tracial Rokhlin property.

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