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A Lichnerowicz Vanishing Theorem for the Maximal Roe Algebra

Hao Guo, Zhizhang Xie, Guoliang Yu

Abstract

We show that if a countable discrete group acts properly and isometrically on a spin manifold of bounded Riemannian geometry and uniformly positive scalar curvature, then, under a suitable condition on the group action, the maximal higher index of the Dirac operator vanishes in K-theory of the maximal equivariant Roe algebra. The group action is not assumed to be cocompact. A key step in the proof is to establish a functional calculus for the Dirac operator in the maximal equivariant uniform Roe algebra. This allows us to prove vanishing of the index of the Dirac operator in K-theory of this algebra, which in turn yields the result for the maximal higher index.

A Lichnerowicz Vanishing Theorem for the Maximal Roe Algebra

Abstract

We show that if a countable discrete group acts properly and isometrically on a spin manifold of bounded Riemannian geometry and uniformly positive scalar curvature, then, under a suitable condition on the group action, the maximal higher index of the Dirac operator vanishes in K-theory of the maximal equivariant Roe algebra. The group action is not assumed to be cocompact. A key step in the proof is to establish a functional calculus for the Dirac operator in the maximal equivariant uniform Roe algebra. This allows us to prove vanishing of the index of the Dirac operator in K-theory of this algebra, which in turn yields the result for the maximal higher index.

Paper Structure

This paper contains 18 sections, 19 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Maximal equivariant Roe algebras
  3. Notation
  4. Geometric conditions
  5. Bounded geometry
  6. A condition on the $\Gamma$-action
  7. Operator algebras
  8. Estimating the maximal norm
  9. Discretizing $\mathcal{H}_M$
  10. Norm estimation
  11. Maximal Roe algebras
  12. Functional calculus
  13. A Hilbert module operator
  14. Regularity and essential self-adjointness
  15. Proof of the main theorem
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $(M^n,g)$ be a Riemannian manifold with bounded Riemannian geometry and $\Gamma$ a countable discrete group acting properly and isometrically on $M$, satisfying Assumption ass:condition. Suppose that $M$ has a $\Gamma$-equivariant spin structure, with $S$ and $D$ the spinor bundle and Dirac oper where $C^*_{\textnormal{max}}(M)^\Gamma$ is the maximal equivariant Roe algebra of $M$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Corollary 2.7
  • Corollary 2.8
  • proof
  • Definition 2.9
  • ...and 33 more