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A higher index and rapidly decaying kernels

Hao Guo, Peter Hochs, Hang Wang

TL;DR

We construct a higher index for an $H$-equivariant, first-order, self-adjoint elliptic operator $D$ in the $K$-theory of the Fréchet algebra $\mathcal{A}_S(E)$ consisting of smooth kernels with faster-than-exponential decay; the index is represented by an idempotent built from heat operators $e^{-tD^2}$ and related heat calculus. This enables pairings with traces and cyclic cocycles via heat-kernel asymptotics and ties to the Roe algebra. The framework provides a bridge to von Neumann algebras and $L^2$-index theorems and includes an equivariant multiplier theory for $D$ and the heat operator. In particular, for cofinite-volume group actions, it yields an explicit $L^2$-index formula for twisted Spin$^c$ Dirac operators: $(\tau_e^{\Gamma} \otimes \mathrm{Tr}_F)(j_{\mathcal{N}}(\mathrm{index}_{\mathbb{C},\Gamma}(D))) = \mathrm{vol}(\Gamma\backslash H) \int_F \hat{A}(X) e^{c_1(L_{\det})/2} \mathrm{ch}(E).$

Abstract

We construct an index of first-order, self-adjoint, elliptic differential operators in the $K$-theory of a Fréchet 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.

A higher index and rapidly decaying kernels

TL;DR

We construct a higher index for an -equivariant, first-order, self-adjoint elliptic operator in the -theory of the Fréchet algebra consisting of smooth kernels with faster-than-exponential decay; the index is represented by an idempotent built from heat operators and related heat calculus. This enables pairings with traces and cyclic cocycles via heat-kernel asymptotics and ties to the Roe algebra. The framework provides a bridge to von Neumann algebras and -index theorems and includes an equivariant multiplier theory for and the heat operator. In particular, for cofinite-volume group actions, it yields an explicit -index formula for twisted Spin Dirac operators:

Abstract

We construct an index of first-order, self-adjoint, elliptic differential operators in the -theory of a Fréchet 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 -index theorems as an immediate application, and work out further applications in other papers.
Paper Structure (10 sections, 21 theorems, 129 equations)

This paper contains 10 sections, 21 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. A higher index
  3. An algebra of exponentially decaying kernels
  4. Construction of the index
  5. Exponentially decaying kernels
  6. The algebra $\mathcal{A}_S(E)$
  7. Suprema of Schwartz kernels
  8. Decay behaviour of Schwartz kernels
  9. Multipliers of $\mathcal{A}_{S, \varphi}(E)^H$
  10. Von Neumann algebras

Key Result

Lemma 2.5

The space $\mathcal{A}_S(E)$ is a Fréchet algebra under the composition eq compos kernels.

Theorems & Definitions (56)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Definition 2.8
  • Example 2.9
  • Example 2.10
  • ...and 46 more