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 criterion for C⁎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 [8]. Several matrix systems studied by Farenick and Paulsen [13] are shown to be test systems for nuclearity. We also prove that the standard Namioka–Phelps test space is C⁎C⁎-nuclear. We propose a partition of unity property for C⁎C⁎-algebras which distinguishes nuclear C⁎C⁎-algebras among the others.