Spectral flow for nonunital spectral triples

TL;DR

This paper advances nonunital index theory by proving spectral triples from real actions satisfy index formula hypotheses and clarifies spectral flow interpretation in nonunital contexts.

Contribution

It establishes conditions under which nonunital spectral triples meet index formula criteria and links spectral flow to earlier analytic definitions in nonunital cases.

Findings

Spectral triples from real actions satisfy nonunital index formula hypotheses.

Spectral flow in nonunital cases can be connected to earlier analytic approaches.

Provides new insights into nonunital index theory and spectral flow interpretation.

Abstract

We prove two results about nonunital index theory left open by [CGRS2]. The first is that the spectral triple arising from an action of the reals on a C*-algebra with invariant trace satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting we are able to connect with earlier approaches to the analytic definition of spectral flow.