Given a C-algebra A with a KMS weight for a circle action, we construct
and compute a secondary invariant on the equivariant K-theory of the mapping cone of
AT ,! A, both in terms of equivariant KK-theory and in terms of a semifinite spectral
flow. This in particular puts the previously considered examples of Cuntz algebras 
and SUqð2Þ  in a general framework. As a new example we consider the Araki–Woods
IIIl representations of the Fermion algebra.