The Possible Temperatures for Flows on a Simple AF Algebra

Klaus Thomsen

doi:10.1007/s00220-021-04130-x

Abstract

It is shown that for any infinite dimensional simple unital AF algebra A and any closed lower bounded set K of real numbers containing zero there is a flow on A for which the set of possible inverse temperatures is K.

Highlights

A flow on a C∗-algebra A is a continuous one-parameter group α =t∈R of automorphisms of A
The C∗algebra A of observables in such a model is often a UHF algebra and it is of interest to determine the KMS states and the possible inverse temperatures which can occur for flows on a UHF algebra
It follows from work by Powers and Sakai in the 70’s, [PS], that when the flow is approximately inner the set of possible inverse temperatures is the whole real line R, and Powers and Sakai conjectured that all flows on a UHF algebra are approximately inner

Summary

Introduction

We will show that a variant of the method developed in [Th5] can be used to construct a flow on any infinite dimensional simple unital AF algebra such that the set of inverse temperatures is any given lower bounded closed set of real numbers containing zero. For this we use a method which is a Version: May 19, 2021. While it is certainly premature to suggest that this applies to all infinite dimensional unital simple C∗-algebras, it does seem appropriate to point out that we don’t really know

Statement of results

Preparations