Paschke Category, K-homology, and the Riemann-Roch Transformation

For a separable C*-algebra A, We define an exact C*-category called the Paschke Category of A, and show that its topological K-theory groups are equal to topological K-homology groups of the C*-algebra A. Then we use the Dolbeault complex and ideas from the classical methods in Kasparov K-theory to construct an acyclic chain complex in this category, which in turn, induces a Riemann-Roch transformation from algebraic K-theory spectra of a complex manifold X, to its topological K-homology spectra. We examine the question of whether this map commutes with push-forward with respect to a proper map of complex manifolds, and how we can extend it to complex spaces.