Classification of actions on the Jiang-Su algebras and their applications

2014 Fiscal Year Final Research Report

• Project/Area Number: 25887031
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Kyoto University
• Principal Investigator: SATO Yasuhiko 京都大学, 理学(系)研究科(研究院), 助教 (70581502)
• Project Period (FY): 2013-08-30 – 2015-03-31
• Keywords: C* 環 / 力学系 / Jiang-Su 環 / 分類理論 / Toms-Winter 予想 / Rosenberg 予想 / Powers-Sakai 予想

Outline of Final Research Achievements

In this study, the following theorems are established. (i) affirmative answer for the Toms-Winter conjecture in the case that the extreme trace space has finite topological covering dimension, (ii) affirmative answer of the Rosenberg conjecture for all elementary amenable groups, (iii) the solution of the Powers-Sakai conjecture.

On the first stage of this study, although we aimed at a result just for the Jiang-Su algebra which plays a central role in the classification of operator algebras, we have success in a consequence of (i) which is a well known open problem concerns with the whole class of classifiable operator algebras. As an application of dynamical systems of this study, we also obtained a partial answer (ii) to the famous open problem. In (iii), a counter-example to the Powers-Sakai conjecture is constructed as the second application. This conjecture indicated the possibility that a physical theory occurs on any dynamical system on any uniformly hyperfinite algebra.

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