Extremal Marginal Tracial States in Coupled Systems

Let Γ be the convex set consisting of all states φ on the tensor product B ⊗ B of the algebra B = Mn(C) of all n × n matrices over the complex numbers C with the property that the restrictions φ B⊗I and φ I⊗B are the unique tracial states on B ⊗ I and I ⊗ B . We find necessary and sufficient conditions for such a state, called a marginal tracial state, to be extremal in Γ . We also give a characterization of those extreme points in Γ which are pure states. We conjecture that all extremal marginal tracial states are pure states. Mathematics subject classification (2000): 46L30, 46L06.