  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.
