Completely positive linear operators for Banach spaces (Q1321843)

Let \(X\) be a Banach space and \(\overline {X^*}\) the set of all conjugate linear functionals on it. An operator \(T\in B(X, \overline {X^*})\) is called here positive, if \[ \sum_{i=1}^ n\;\sum_{j=1}^ n T(x_ i) (x_ j)\geq 0, \qquad \text{for all} \quad n\in\mathbb{N},\;x_ i\in X. \] Using this concept, the author defines, in a standard way, completely positive linear maps \(\varphi: A\to B(X, \overline{X^*})\), where \(A\) is a \(C^*\)-algebra, and proves a Stinespring type representation theorem for such maps. The paper also contains analogues of Arveson's extreme point results in this more general setting.