On a non-commutative analogue of a classical result of Namioka and Phelps

A classical result of Namioka and Phelps states that the square is a test object to verify semi-simplexity in the tensor theory of convex compact sets. By using the quantization of generalized Namioka-Phelps test spaces we formulate a nuclearity criteria for C*-algebras, which establishes a non-commutative version of their result. The proof we suggest covers the nuclearity characterization via non-commutative polyhedron outlined by Effros. Several matrix systems studied by Farenick and Paulsen are shown to be test systems for nuclearity. We also prove that the standard Namioka-Phelps test space is C*-nuclear. We propose a partition of unity property for C*-algebras which distinguishes nuclear C*-algebras among the others.