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Uniqueness of extremal almost periodic states on the injective type III$_{1}$ factor
- Publication Year :
- 2024
-
Abstract
- Let $R_\infty$ denote the Araki--Woods factor -- the unique separable injective type III$_{1}$ factor. For extremal almost periodic states $\varphi, \psi\in (R_\infty)_*$, we show that if $\Delta_\varphi$ and $\Delta_\psi$ have the same point spectrum then $\psi = \varphi\circ \alpha$ for some $\alpha\in $ Aut$(R_\infty)$. Consequently, the extremal almost periodic states on $R_\infty$ are parameterized by countable dense subgroups of $\mathbb{R}_+$, up to precomposition by automorphisms. As an application, we show that KMS states for generalized gauge actions on Cuntz algebras agree (up to an automorphism) with tensor products of Powers states on their von Neumann completions.<br />Comment: 18 pages
- Subjects :
- Mathematics - Operator Algebras
46L30
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2406.00874
- Document Type :
- Working Paper