C*-independence, product states and commutation

Let D be a unital C*-algebra generated by C*-subalgebras A and B possessing the unit of D. Motivated by the commutation problem of G*-independent, algebras arising in quantum field theory, the interplay between commutation phenomena, product type extensions of pairs of states and tensor product, structure is studied. Roos's theorem [11] is generalized in showing that the following conditions are equivalent: (i) every pair of states on A and B extends to an uncoupled product state on D; (ii) there is a representation pi of D such that pi(A) and pi(B) commute and pi is faithful on both A and B; (iii) A circle times(min) B is canonically isomorphic to a quotient of D. The main results involve unique common extensions of pairs of states. One consequence of a general theorem proved is that, in conjunction with the unique product state extension property, the existence of a faithful family of product states forces commutation. Another is that if D is simple and has the unique product extension property across A and B then the latter C*-algebras must, commute and D be their minimal tensor product.