Take a unital cp map $f:B\to A$ between unital $C^*$ algebras. Given a state $\psi:B\to \mathbb{C}$ what conditions are necessary for there to exist a state $\phi:A\to \mathbb{C}$ so that $\phi\circ f=\psi$? I am sure that the answer must be known, and apologise for my ignorance.

In the case where $f$ is a unital $*$-algebra hom, and we consider the image of $f$ as a subalgebra, suitable conditions are given in -- Joel Anderson, Extensions, restrictions, and representations of states on $C^*$-algebras, Transactions of the American Mathematical Society 249(2):303-329, 1979.

This comes from considering a possibility of defining the degree of a cp map at a pure state. The classical theory requires taking the inverse image of points, which translates to the current question. The states may be assumed pure if it helps (as in the paper above).

As Nik points out below, the Hahn Banach theorem proves this if we have the inequality |ψ(b)|≤‖f(b)‖ for all b∈B.

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