Type III_1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers

TL;DR

This paper completes the classification of KMS-states for the Toeplitz algebra of the affine semigroup over natural numbers, showing that for certain temperatures, these states are of type III_1, linking to the Bost-Connes system.

Contribution

It provides a full analysis of the type classification of KMS-states in the Toeplitz algebra, connecting it to the Bost-Connes system and parametrizing the Nica spectrum via adelic spaces.

Findings

KMS_beta-states are of type III_1 for beta in [1,2]

Parametrization of the Nica spectrum using adelic spaces

Recovery of all previously computed KMS-states

Abstract

We complete the analysis of KMS-states of the Toeplitz algebra of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature beta in the critical interval [1,2], the unique KMS_beta-state is of type III_1. We prove this by reducing the type classification from the Toeplitz algebra to that of the symmetric part of the Bost-Connes system, with a shift in inverse temperature. To carry out this reduction we first obtain a parametrization of the Nica spectrum of the Toeplitz algebra in terms of an adelic space. Combining a characterization of traces on crossed products due to the second author with an analysis of the action of the affine semigroup on the Nica spectrum, we can also recover all the KMS-states originally computed by Raeburn and the first author. Our computation sheds light on why there is a free transitive circle action on the extremal KMS_beta-states for beta>2 that does not ostensibly come from an action on the C*-algebra.