Equivalent conditions for the general Stone-Weierstrass problem (Q1067179)

The setting of the Stone-Weierstraß\ problem is generalized to the cone \(CP(A,H)\) of all completely positive linear maps of a unital \(C^*\)-algebras \(A\) into the \(C^*\)-algebra \(B(H)\) of all bounded linear operators on a Hilbert space \(H\). It is shown that some conditions are equivalent to that of Stone-Weierstraß\ conjecture, i.e., a \(C^*\)-subalgebra \(B\) of \(A\) which contains the identity of \(A\) separates the pure states of \(A\). One of such conditions is that \(B\) separates the pure completely positive linear maps \(P_ A(H)\) of \(A\) into \(B(H)\). As a consequence, a new generalized spectrum ``\(\oplus_{\dim H\leq \alpha_ A}P_ A(H)\)'' is introduced, where \(\alpha_ A=\sup_{\pi \in \hat A}\dim \pi\) and \(\hat A\) is the set of all unitary equivalence classes of the set of irreducible representations. This generalized spectrum contains pure states and irreducible representations as proper subsets.