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 of N⋊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 of N⋊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.