Spectral flow and the unbounded Kasparov product

TL;DR

This paper develops a general method for constructing unbounded representatives of the interior Kasparov product, utilizing C^1-connections on operator modules without smoothness assumptions, and applies it to spectral flow and index theory of Dirac operators.

Contribution

It introduces a new framework for unbounded Kasparov products using C^1-connections, broadening the applicability of index theory in operator algebras.

Findings

Constructed unbounded representatives for the interior Kasparov product.

Connected spectral flow and index theory to Kasparov product.

Provided minimal assumptions for operator module connections.

Abstract

We present a fairly general construction of unbounded representatives for the interior Kasparov product. As a main tool we develop a theory of C^1-connections on operator * modules; we do not require any smoothness assumptions; our sigma-unitality assumptions are minimal. Furthermore, we use work of Kucerovsky and our recent Local Global Principle for regular operators in Hilbert C*-modules. As an application we show that the Spectral Flow Theorem and more generally the index theory of Dirac-Schr\"odinger operators can be nicely explained in terms of the interior Kasparov product.