Completely bounded maps on \(C^*\)-algebras and invariant operator ranges. (English) Zbl 0554.46028

Paulsen, Vern I.

Proc. Am. Math. Soc. 86, 91-96 (1982).

Wittstock used deep results on \(W^*\)-algebras and a Hahn-Banach theorem for set-valued matrix sublinear functionals to prove that the completely positive maps from a \(C^*\)-algebra into L(H) span the completely bonded maps. The author presents an elementary proof of this based on Arveson’s extension theorem. He also derives an equivalent reformulation of the invariant operator range problem.

Reviewer: P.G.Spain

MSC:

46L05	General theory of \(C^*\)-algebras
47A15	Invariant subspaces of linear operators

Keywords:

the completely positive maps from a \(C^*\)-algebra into L(H) span the completely bonded maps; Arveson’s extension theorem; invariant operator range problem