  We initiate a study of linear maps on $M_n(\mathbb{C})$ that have the
property that they factor through a tracial von Neumann algebra
$(\mathcal{A,\tau})$ via operators $Z\in M_n(\mathcal{A})$ whose entries
consist of positive elements from the von-Neumann algebra. These maps often
arise in the context of non-local games, especially in the synchronous case. We
establish a connection with the convex sets in $\mathbb{R}^n$ containing
self-dual cones and the existence of these maps. The Choi matrix of a map of
this kind which factors through an abelian von-Neumann algebra turns out to be
a completely positive (CP) matrix.
  We fully characterize positively factorizable maps whose Choi rank is 2. We
also provide some applications of this analysis in finding doubly nonnegative
matrices which are not CPSD. A special class of these examples is found from
the concept of Unextendible Product Bases in quantum information theory.
