Completely positive invariant conjugate-bilinear maps in partial *-algebras (Q2460708)

The classical Stinespring dilation theorem says that a linear map \(T: A\to B(H)\), where \(A\) is a unital \(C^*\)-algebra and \(B(H)\) is the \(C^*\)-algebra of bounded operators on the Hilbert space \(H\), is completely positive if and only if it has the form \(T(a)= V^*\pi(a)V\), \(a\in A\), where \(\pi\) is a bounded representation of \(A\) in the Hilbert space \(K\) and \(V: H\to K\) is a bounded linear map. In the present paper, the authors study the possibility of extending this result to the very general setting of partial \(*\)-algebras \(A\) in the place of \(C^*\)-algebras. As it is impossible to keep the usual notion of completely positive linear maps in this case, the authors consider instead completely positive conjugate bilinear maps defined on a subspace of \(A\times A\). Under certain additional conditions, the authors define ``Stinespring dilations'' and prove a generalized Stinespring theorem.