Centre-valued index for Toeplitz operators with noncommuting symbols

TL;DR

This paper introduces a centre-valued index theory for Toeplitz operators with noncommuting symbols, extending previous scalar-valued results to a more general setting with invariant traces.

Contribution

It defines a centre-valued winding number and provides an index formula for Toeplitz operators with noncommuting symbols, generalizing earlier scalar-valued trace results.

Findings

Established a centre-valued index formula for Toeplitz operators

Defined a centre-valued winding number using the trace

Extended previous scalar-valued trace results to noncommuting symbols

Abstract

We consider an action of the real line on a C*-algebra for which there is a centre-valued invariant trace. We define a family of Toeplitz operators with symbols in the original algebra. When the symbol is invertible, the Toeplitz operator is Fredholm in an appropriate sense, and we give a formula for the index using a notion of centre-valued winding number defined using the trace. The results and techniques generalise previous results of the authors for scalar-valued traces.