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Equivalent conditions for the general Stone-Weierstrass problem. (English) Zbl 0579.46040

The setting of the Stone-Weierstraßproblem 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-Weierstraßconjecture, 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.
Reviewer: Ichiro Fujimoto

MSC:

46L05 General theory of \(C^*\)-algebras
46L30 States of selfadjoint operator algebras
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References:

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