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Stinespring representability and Kadison’s Schwarz inequality in non-unital Banach star algebras and applications. (English) Zbl 0938.46049

The author studies the Stinespring representability of completely positive linear map of Banach *-algebra to the C * -algebra B(H) of all bounded linear operators on a Hilbert space H.

Let (A,) be a complex Banach *-algebra with x * =x, xA, not necessarily having an identity, and ϕ a completely positive linear map of A to B(H). He obtains that the following conditions (1)-(6) are equivalent:

(1) ϕ is Stinespring representable, that is, there exist a Hilbert space K, a *-homomorphism π:AB(H) and a bounded linear operator V:HK such that ϕ(x)=V * π(x)V, xA and K is the closed linear span of π(A)VH.

(2) There exists a constant k>0 such that ϕ(x) * ϕ(x)kϕ(x * x), xA.

(3) ϕ(x) * =ϕ(x * ), xA and there exists a constant k>0 such that ϕ(h) 2 kϕ(h 2 ) for all h * =hA.

(4) ϕ is extendable to a completely positive linear map ϕ e on the unitization A e of A to B(H).

(5) ϕ is continuous in the Gelfand-Naimark pseudo-norm p .

(6) There exists a completely positive linear map ϕ ˜ of the enveloping C * -algebra C * (A) of A to B(H) such that ϕ=ϕ ˜j, where j:xAx+Kerp C * (A).

Furthermore, he obtains a similar result for positive linear maps.

MSC:
46K10Representations of topological algebras with involution
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