Summary: Let be a unital -algebra generated by -subalgebras and possessing the unit of . Motivated by the commutation problem of -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 [H. Roos, Commun. Math. Phys. 16, 238–246 (1970; Zbl 0197.26303)] is generalized in showing that the following conditions are equivalent: (i) every pair of states on and extends to an uncoupled product state on ; (ii) there is a representation of such that and commute and is faithful on both and ; (iii) is canonically isomorphic to a quotient of .
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 is simple and has the unique product extension property across and then the latter -algebras must commute and be their minimal tensor product.