The Index of Toeplitz Operators on Free Transformation Group C*-Algebras

Let be a discrete group acting on a compact manifold X, letV be a -equivalent Hermitian vector bundle over X, and let Dbe a first-order elliptic self-adjoint -equivalent differentialoperator acting on sections of V. This data is used to defineToeplitz operators with symbols in the transformation groupC*-algebra C(X), and it is shown that if the symbol of sucha Toeplitz operator is invertible, then the operator is Fredholm.In the case where is finite and acts freely on X, a geometric-topologicalformula for the index is stated that involves an explicitlyconstructed differential form associated to the symbol. 2000Mathematics Subject Classification 47A53 (primary), 19K56, 47B35,46L87 (secondary).