Selfless C*-correspondences, operator valued C*-probability spaces and completely positive maps

We develop a general theory of selflessness for C*-correspondences, with several applications.

Specializing this theory to completely positive maps, in particular to conditional expectations, gives rise to a novel notion of relative selflessness.

Among applications, the machinery developed here yields new examples of selfless C*-algebras, for instance among minimal tensor products and reduced crossed products, new examples of MF C*-algebras arising as reduced amalgamated free products, a conceptually new proof of Kirchberg's $\mathcal{O}_{\infty}$-absorption theorem, and the lack of ``phantom'' traces on ultrapowers.