Every interior point of an indecomposable homogeneous cone admits a purification in the maximal tensor product, with uniqueness controlled by extreme positive maps fixing the unit and local automorphisms.
Notes on extremality of the Choi map.Linear Algebra and its Applications, 439 (10):3156–3165, 2013
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Purifications for Convex Cones
Every interior point of an indecomposable homogeneous cone admits a purification in the maximal tensor product, with uniqueness controlled by extreme positive maps fixing the unit and local automorphisms.