Index theory of Toeplitz operators associated to transformation group C^∗-algebras

Efton Park

Vol. 223 (2006), No. 1, 159–165

DOI: 10.2140/pjm.2006.223.159

Abstract

Let Γ be a finite discrete group acting smoothly on a compact manifold X, and let D be a first-order elliptic self-adjoint Γ-equivariant differential operator acting on sections of some Γ-equivariant Hermitian vector bundle over X. We use these data to define Toeplitz operators with symbols in the transformation group C^∗-algebra C(X) ⋊ Γ. If the symbol of such a Toeplitz operator is invertible, then the operator is Fredholm. In the case where X is a spin manifold and D is the Dirac operator, we give a geometric-topological formula for the index.

Keywords

Toeplitz operators, index theory, transformation group C^∗-algebras

Mathematical Subject Classification 2000

Primary: 47A53

Secondary: 19K56, 47B35, 46L87

Milestones

Received: 10 December 2003

Revised: 22 July 2005

Accepted: 23 July 2005

Published: 1 January 2006