The general Stone-Weierstrass problem and extremal completely positive maps (Q1820366)

The general Stone-Weierstrass conjecture for unital \(C^*\)-algebras \(B\subseteq A\) states that \(A=B\) if B separates the pure states of A. The author proves that the condition of B separating the pure states of A is equivalent to B separating the set of extremal completely positive unital maps of A into L(H) for any finite dimensional Hilbert space H. Furthermore he proves that \(B=A\) if this condition holds for any Hilbert space H with dim \(H\leq card(S)\), the cardinality of some dense subset S of A.