Factorization properties for unbounded local positive maps

TL;DR

This paper investigates the factorization properties of unbounded local positive maps in the context of locally C*-algebras, establishing conditions under which these maps can be decomposed into tensor products of simpler maps.

Contribution

It introduces new factorization results for unbounded local positive maps and characterizes when such maps are local decomposable or can be expressed as tensor products.

Findings

Unbounded local positive maps dominated by a tensor product can be factorized into simpler tensor components.

Local positive maps tensorized with identity are local decomposable iff they are local CP-maps.

Unbounded local CCP-maps dominated by tensor products can be expressed as tensor products of unbounded local CCP-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.