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 $(\mathcal{A},\mathit{\tau })$ 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 factor 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 are found from the concept of Unextendible Product Bases in quantum information theory.

我们开始研究线性地图 ${米}_n(\mathbb{C})$ 具有通过迹冯诺依曼代数分解的性质 $(\mathcal{一种},\mathit{\tau })$ 通过运营商 $Z\in {米}_n(\mathcal{一种})$其条目由冯诺依曼代数中的正元素组成。这些地图通常出现在非本地游戏的上下文中，尤其是在同步情况下。我们建立与凸集的连接${\mathbb{电阻}}^n$包含自对偶锥体和这些映射的存在。这种映射的 Choi 矩阵通过阿贝尔 von-Neumann 代数因子证明是完全正 (CP) 矩阵。我们充分表征了 Choi 秩为 2 的可正分解图。我们还提供了这种分析在寻找不是 CPSD 的双重非负矩阵方面的一些应用。从量子信息理论中不可扩展的产品基础的概念中可以找到这些例子中的一类特殊。