Stinespring representability and Kadison's Schwarz inequality in non-unital Banach star algebras and applications (Q1304599)

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 \(\|x^*\|=\|x\|\), \(x\in A\), not necessarily having an identity, and \(\phi\) a completely positive linear map of \(A\) to \(B(H)\). He obtains that the following conditions \((1)-(6)\) are equivalent: (1) \(\phi\) is Stinespring representable, that is, there exist a Hilbert space \(K\), a \(*\)-homomorphism \(\pi:A\to B(H)\) and a bounded linear operator \(V:H\to K\) such that \(\phi(x)= V^*\pi(x) V\), \(x\in A\) and \(K\) is the closed linear span of \(\pi(A)VH\). (2) There exists a constant \(k>0\) such that \(\phi(x)^*\phi(x)\leq k\phi(x^*x)\), \(\forall x\in A\). (3) \(\phi(x)^*=\phi(x^*)\), \(^{\forall}x\in A\) and there exists a constant \(k>0\) such that \(\phi(h)^2\leq k\phi(h^2)\) 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 \(B(H)\). (5) \(\phi\) is continuous in the Gelfand-Naimark pseudo-norm \(p_\infty\). (6) There exists a completely positive linear map \(\widetilde\phi\) of the enveloping \(C^*\)-algebra \(C^*(A)\) of \(A\) to \(B(H)\) such that \(\phi=\widetilde\phi\circ j\), where \(j:x\in A\mapsto x+\text{Ker }p_\infty\in C^*(A)\). Furthermore, he obtains a similar result for positive linear maps.