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 $\varphi$ 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 Fréchet quantized domain, that is dominated by $\phi \otimes$id is of the forma $\psi \otimes$id, where $\psi$ is an unbounded local positive map dominated by $\phi$. As an application of this result, we show that given a local positive map $\phi :$ $\mathcal{A}\to$ $\mathcal{B},$ the local positive map $\phi \otimes$id${}_{M_n\left(\mathbb{C}\right)}$ is local decomposable for some $n\ge 2$ if and only if $\phi$ is a local $CP$-map. Also, we show that an unbounded local $CCP$-map $\varphi$ on the minimal tensor product of the unital locally $C^{\ast }$-algebras $\mathcal{A}$ and $\mathcal{B},$ that is dominated by $\phi \otimes \psi$ is of the forma $\phi \otimes \tilde{\psi }$, where $\tilde{\psi }$ is an unbounded local $CCP$- map dominated by $\psi$, whenever $\phi$ is pure.

References in corpus (1)

• Operator Models for Hilbert Locally $C^{\ast }$-Modules