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

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