Index Theory of Toeplitz Operators on Transformation Group C*-algebras
by
Efton Park
Texas Christian University

Let X be a compact manifold, let E be a Hermitian vector bundle over X, and let D be a first-order elliptic self-adjoint differential operator acting on L²-sections L²(E) of E. Let P be the positive spectral projection of D, and for each f in C(X), let f act on L²(E) by pointwise multiplication. Then it is well-known that if f is nowhere-vanishing, the Toeplitz operator T_f = Pf is a Fredholm operator on the range of P. Furthermore, there is a geometric-topological formula for the index of T_f.

Now suppose that a discrete group G acts on both X and E and that D commutes with the action of G on sections of E. Then the crossed-product of C(X) by G acts on L²(E), and if F is an invertible element of this algebra, the operator T_F = PF is a Fredholm operator on the range of P. In this talk, I will discuss the properties of these Toeplitz operators, and for certain cases I will give a geometric-topological formula for the Fredholm index of such operators.

Date received: March 5, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacw-03.