Coronas and Callias type operators in coarse geometry

TL;DR

This paper explores the relationship between Callias type Dirac operators and coarse geometry, interpreting their symbols and index classes through pairings with K-theory classes of the coarse corona, incorporating positivity and invertibility conditions.

Contribution

It introduces a novel interpretation of the coarse symbol and index class of Callias type operators as pairings with K-theory classes in coarse geometry, linking operator theory with geometric structures.

Findings

Provides a K-theoretic framework for Callias operators in coarse geometry

Establishes support conditions for positivity and invertibility

Connects local operator properties with coarse geometric invariants

Abstract

We interpret the coarse symbol and index class of a Callias type Dirac operator $D+\Psi$ on a manifold $M$ as a pairing between the coarse symbol and index classes associated to $D$ and K-theory classes of the coarse corona of $M$ or $M$ itself determined by $\Psi$. Local positivity of $D$ and local invertibility of $\Psi$ are incorporated in terms of support conditions on the $K$-theoretic level.