This question is about mapping finitely generated projective modules over $C^*$ algebras (or K-theory classes - I don't mind much) under maps between algebras. If we have an idempotent matrix $P\in M_n(A)$ and a star algebra map $\phi:A\to B$ then taking $\phi(P)$ (applying to each entry) will give another idempotent. However suppose that we only have a completely positive map $\psi:A\to B$ -- what do we do then?

One possible way would be to use the KSGNS theorem to say that $\psi$ is given by a Hilbert bimodule structure on some $M$ together with an element $m\in M$. The module given by the applying the map to an $A$-module $E$ might be taken as $M\otimes_A E$ (or rather the conjugate of $M$ to get the sides right...). However this seems to be a very non-unique construction, and likely far too big.

The reason I am interested in this is the problem of calculating characteristic classes (a la Connes) for algebras which simply do not have any nicely calculable $n$-cycles. It may be simplest to transfer the bundle to an easier to deal with algebra. (Just as classically we can define a cycle on a manifold by using an embedded submanifold). However then we run into the problem that we often only have CP maps, not algebra maps... I would be grateful for any assistance.