Twisted cyclic theory, equivariant KK theory and KMS States

TL;DR

This paper develops a general framework connecting KMS states, twisted cyclic cocycles, and equivariant KK-theory for circle actions on C*-algebras, extending previous index theory results and introducing new eta cocycles.

Contribution

It introduces a unified approach to index theory for KMS states using twisted cyclic cocycles and equivariant KK-theory, including the discovery of eta cocycles and new examples beyond Cuntz-Krieger systems.

Findings

Constructed Kasparov modules and spectral triples from KMS states.

Associated twisted cyclic cocycles with these triples via a residue construction.

Identified examples using Fermion algebra representations that are not Cuntz-Krieger systems.

Abstract

Recently, examples of an index theory for KMS states of circle actions were discovered, \cite{CPR2,CRT}. We show that these examples are not isolated. Rather there is a general framework in which we use KMS states for circle actions on a C*-algebra A to construct Kasparov modules and semifinite spectral triples. By using a residue construction analogous to that used in the semifinite local index formula we associate to these triples a twisted cyclic cocycle on a dense subalgebra of A. This cocycle pairs with the equivariant KK-theory of the mapping cone algebra for the inclusion of the fixed point algebra of the circle action in A. The pairing is expressed in terms of spectral flow between a pair of unbounded self adjoint operators that are Fredholm in the semifinite sense. A novel aspect of our work is the discovery of an eta cocycle that forms a part of our twisted residue cocycle. To illustrate our theorems we observe firstly that they incorporate the results in \cite{CPR2,CRT} as special cases. Next we use the Araki-Woods III_\lambda representations of the Fermion algebra to show that there are examples which are not Cuntz-Krieger systems.