Supersymmetry and Fredholm modules over quantized spaces

The purpose of this paper is to apply the framework of non-commutative differential geometry to quantum deformations of a class of Kähler manifolds. For the examples of the Cartan domains of type I and flat space, we construct Fredholm modules over the quantized manifolds using the supercharges which arise in the quantization of supersymmetric generalizations of the manifolds. We compute an explicit formula for the Chern character on generators of the Toeplitz ℂ^*-algebra.