Narrow Results

We apply Hilbert module methods to show that normal completely positive maps admit weak tensor dilations. Appealing to a duality between weak tensor dilations and extensions of CP-maps, we get an existence proof for certain extensions. We point out that this duality is part of a far reaching duality between a von Neumann bimodule and its commutant in which other dualities, known and new, also find their natural common place.

The purpose of this short paper is to clarify and present a general version of an interesting observation by [Piani and Mora, Phys. Rev. A 75 (2007) 012305], linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let $A_i$, $C_i$ be unital C*-algebras and let ${\alpha }_i$ be positive linear maps from $A_i$ to $C_i,$$i=1,2$. We obtain conditions under which any positive map $\beta$ from the minimal C*-tensor product $A_1{\otimes }_{min}{A}_2$ to $C_1{\otimes }_{min}{C}_2$, such that ${\alpha }_1\otimes {\alpha }_2\ge \beta$, factorizes as $\beta =\gamma \otimes {\alpha }_2$ for some positive map $\gamma$. In particular, we show that when ${\alpha }_i:A_i\to B({{\mathscr{H}}}_i)$ are completely positive (CP) maps for some Hilbert spaces ${{\mathscr{H}}}_i$$(i=1,2)$, and ${\alpha }_2$ is a pure CP map and $\beta$ is a CP map so that ${\alpha }_1\otimes {\alpha }_2-\beta$ is also CP, then $\beta =\gamma \otimes {\alpha }_2$ for some CP map $\gamma$. We show that a similar result holds in the context of positive linear maps when $A_2=C_2=B({\mathscr{H}})$ and ${\alpha }_2=id$. As an application, we extend IX Theorem of Ref. 4 (revisited recently by [Huber et al., Phys. Rev. Lett. 121 (2018) 200503]) to show that for any linear map $\tau$ from a unital C*-algebra $A$ to a C*-algebra $C$, if $\tau \otimes i{d}_k$ is decomposable for some $k\ge 2$, where $i{d}_k$ is the identity map on the algebra $M_k(ℂ)$ of $k\times k$ matrices, then $\tau$ is CP.