Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras

Understand

This paper presents, by example, an index theory appropriate to algebras without trace.

• Whilst we work exclusively with the 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 the Cuntz algebra.
• We introduce a modified $K_1$-group of the Cuntz algebra so as to pair with this twisted cocycle.