Twisted cyclic theory and an index theory for the gauge invariant KMS state on the Cuntz algebra O n

This paper presents, by example, an index theory appropriate to algebras without trace. Whilst we work exclusively with Cuntz algebras the exposition is designed to indicate how to develop a general theory. Our main result is an index theorem (formulated in terms of spectral flow) using a twisted cyclic cocycle where the twisting comes from the modular automorphism group for the canonical gauge action on each Cuntz algebra. We introduce a modified K 1 -group for each Cuntz algebra which has an index pairing with this twisted cocycle. This index pairing for Cuntz algebras has an interpretation in terms of Araki's notion of relative entropy.