Let Γ be a discrete group acting on a compact manifold X, let V be a Γ-equivalent Hermitian vector
bundle over X, and let D be a first-order elliptic self-adjoint Γ-equivalent 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)[rtimes ]Γ, and it is shown that if the symbol of such a Toeplitz operator is invertible, then the
operator is Fredholm. In the case where Γ 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.
Toeplitz operators and solvable $C^*$-algebras on Hermitian symmetric spaces