A higher index and rapidly decaying kernels

TL;DR

This paper develops a new index theory for elliptic operators with rapidly decaying kernels in a Fréchet algebra, enabling heat kernel methods and applications to von Neumann algebras and L^2-index theorems.

Contribution

It introduces an index in K-theory for operators with super-exponential decay, represented via heat operators, facilitating convergence and computational techniques.

Findings

Constructed an index in K-theory for rapidly decaying kernels.

Represented the index using heat operators for computational advantages.

Linked the theory to von Neumann algebras and L^2-index theorems.

Abstract

We construct an index of first-order, self-adjoint, elliptic differential operators in the $K$-theory of a Fr\'echet algebra of smooth kernels with faster than exponential off-diagonal decay. We show that this index can be represented by an idempotent involving heat operators. The rapid decay of the kernels in the algebra used is helpful in proving convergence of pairings with cyclic cocycles. Representing the index in terms of heat operators allows one to use heat kernel asymptotics to compute such pairings. We give a link to von Neumann algebras and $L^2$-index theorems as an immediate application, and work out further applications in other papers.