The index of Toeplitz operators on free transformation group C*-algebras

Let Gamma be a discrete group acting on a compact manifold X, let V be a Gamma-equivariant Hermitian vector bundle over X, and let D be a first-order elliptic self-adjoint Gamma-equivariant differential operator acting on sections of V. This data is used to define Toeplitz operators with symbols in the transformation group C*-algebra C(X) x Gamma, and it is shown that if the symbol of such a Toeplitz operator is invertible, then the operator is Fredholm. In the case where Gamma is finite and acts freely on X, a geometric-topological formula for the index is stated that involves an explicitly constructed differential form associated to the symbol.