Equivalent conditions for the general Stone-Weierstrass problem

We generalize the setting of the Stone-Weierstrass problem to the cone CP(A, H) of all completely positive linear maps of a C^*-algebra A into the C^*-algebra B(H) of all bounded linear operators on a Hilbert space H, and give some equivalent conditions for the problem. As a consequence, a new generalized spectrum, containing pure states and irreducible representations, will be introduced.