Extremal matrix states on tensor product of C^{*}-algebras

A classic result of Namioka and Phelps on the extreme points of the state space on the tensor product of order unit spaces is generalized to the setting of matrix convexity. We show that the matrix extreme points of the matrix state space on the tensor product of two unital \(C^*\)-algebras, at least one of them is of type \(\mathrm {I}\), are exactly the isometric orbit of the restrictions, on the exponentially unitary orbit of the \(C^*\)-algebraic tensor product, of the tensor products of the matrix extreme points of matrix state spaces of the two \(C^*\)-algebras.