Uniqueness of extremal almost periodic states on the injective type III$_{1}$ factor

TL;DR

This paper characterizes extremal almost periodic states on the unique injective type III$_{1}$ factor, showing they are uniquely determined by their point spectrum up to automorphism, and applies this to KMS states on Cuntz algebras.

Contribution

It establishes a classification of extremal almost periodic states on the Araki--Woods factor based on their spectral data, revealing their parameterization and automorphism invariance.

Findings

Extremal almost periodic states are classified by their point spectrum.

States are parameterized by countable dense subgroups of positive reals.

KMS states on Cuntz algebras are equivalent to tensor powers of Powers states.

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.