We describe a categorical framework for the classification of D-branes on non-commutative spaces using techniques from bivariant K-theory of C*-algebras. We present a new description of bivariant K-theory in terms of non-commutative correspondences which is nicely adapted to the study of T-duality in open string theory. We systematically use the diagram calculus for bivariant K-theory as detailed in our previous paper . We explicitly work out our theory for a number of examples of noncommutative manifolds.
"Non-commutative correspondences, duality and D-branes in bivariant K-theory." Adv. Theor. Math. Phys. 13 (2) 497 - 552, April 2009.