Arveson's extension theorem for conditionally unital completely positive maps (Q6572937)

Motivated by those properties that are satisfied by generators of uniformly continuous semigroups of unital completely positive maps, the author defines and studies the notion of \(\phi\)-conditionally unital completely positive maps. More precisely, given two unital operator systems \(V\) and \(W\), we denote by \(\mathrm{ucp}(V,W)\) the set of all unital completely positive maps from \(V\) to \(W\). For \(\phi\in \mathrm{ucp}(V, W)\), a linear map \(A:V\to W\) is said to be \textit{\(\phi\)-conditionally completely positive} if for each \(n\in \mathbb{N}\), \(v\in M_n(V)\) and \(\rho\in \mathrm{ucp}(M_n(W), \mathbb{C})\) with \(\rho\big(\phi^{(n)}(v)\big) = 0\), one has \(\rho\big(A^{(n)}(v)\big) = 0\). Moreover, a \(\phi\)-conditionally completely positive map \(A\) is called a \textit{\(\phi\)-conditionally unital completely positive map} if \(A(e) = 0\), where \(e\) is the order unit of \(V\).\N\NThe main results of this paper are a form of Choi-Jamiołkowski duality for \(\phi\)-completely positive maps and a form of Arveson's extension theorem for \(\phi\)-conditionally unital completely positive maps.