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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 $B\left(H\right)$ of all bounded linear operators on a Hilbert space $H$.

Let $\left(A,\parallel \phantom{\rule{4pt}{0ex}}\parallel \right)$ be a complex Banach $*$-algebra with $\parallel {x}^{*}\parallel =\parallel x\parallel$, $x\in A$, not necessarily having an identity, and $\phi$ a completely positive linear map of $A$ to $B\left(H\right)$. He obtains that the following conditions $\left(1\right)-\left(6\right)$ are equivalent:

(1) $\phi$ is Stinespring representable, that is, there exist a Hilbert space $K$, a $*$-homomorphism $\pi :A\to B\left(H\right)$ and a bounded linear operator $V:H\to K$ such that $\phi \left(x\right)={V}^{*}\pi \left(x\right)V$, $x\in A$ and $K$ is the closed linear span of $\pi \left(A\right)VH$.

(2) There exists a constant $k>0$ such that $\phi {\left(x\right)}^{*}\phi \left(x\right)\le k\phi \left({x}^{*}x\right)$, $\forall x\in A$.

(3) $\phi {\left(x\right)}^{*}=\phi \left({x}^{*}\right)$, ${}^{\forall }x\in A$ and there exists a constant $k>0$ such that $\phi {\left(h\right)}^{2}\le k\phi \left({h}^{2}\right)$ for all ${h}^{*}=h\in A$.

(4) $\phi$ is extendable to a completely positive linear map ${\phi }^{e}$ on the unitization ${A}_{e}$ of $A$ to $B\left(H\right)$.

(5) $\phi$ is continuous in the Gelfand-Naimark pseudo-norm ${p}_{\infty }$.

(6) There exists a completely positive linear map $\stackrel{˜}{\phi }$ of the enveloping ${C}^{*}$-algebra ${C}^{*}\left(A\right)$ of $A$ to $B\left(H\right)$ such that $\phi =\stackrel{˜}{\phi }\circ j$, where $j:x\in A↦x+\text{Ker}\phantom{\rule{4.pt}{0ex}}{p}_{\infty }\in {C}^{*}\left(A\right)$.

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

MSC:
 46K10 Representations of topological algebras with involution
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