Factorization properties for unbounded local positive maps

Abstract

In this paper we present some factorization properties for unbounded local positive maps. We show that an unbounded local positive map $\phi $ on the minimal tensor product of the locally $C^{\ast }$-algebras $\mathcal{A}$ and $C^{\ast }(\mathcal{D}_{\mathcal{E}}),$ where $\mathcal{D}_{\mathcal{E}}$ is a Fr\'{e}chet quantized domain, that is dominated by $\varphi \otimes $id is of the forma $\psi \otimes $id, where $\psi $ is an unbounded local positive map dominated by $\varphi $. As an application of this result, we show that given a local positive map $\varphi :$ $\mathcal{A}\rightarrow $ $\mathcal{B},$ the local positive map $\varphi \otimes $id$_{M_{n}\left( \mathbb{C}\right) }$ is local decomposable for some $n\geq 2$ if and only if $\varphi $ is a local $CP$-map. Also, we show that an unbounded local $CCP$-map $\phi $ on the minimal tensor product of the unital locally $C^{\ast }$-algebras $\mathcal{A}$ and $\mathcal{B},$ that is dominated by $\varphi \otimes \psi $ is of the forma $\varphi \otimes \widetilde{\psi }$, where $\widetilde{\psi }$ is an unbounded local $CCP$- map dominated by $\psi $, whenever $\varphi $ is pure.