Type III1 equilibrium states of the Toeplitz algebra of the affine semigroup over the natural numbers

We complete the analysis of KMS-states of the Toeplitz algebra T (N N × ) of the affine semigroup over the natural numbers, recently studied by Raeburn and the first author, by showing that for every inverse temperature β in the critical interval 1 β 2, the unique KMSβ-state is of type III1. We prove this by reducing the type classification from T (N N × ) 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 ofN N × 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 ofN N × on the Nica spectrum, we can also recover all the KMS-states of T (N N × ) originally computed by Raeburn and the first author. Our computation sheds light on why there is a free transitive circle action on the extremal KMSβ-states for β >2 that does not ostensibly come from an action of T on the C∗-algebra. © 2011 Elsevier Inc. All rights reserved.