Factorization properties for unbounded local positive maps

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

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