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C * -independence, product states and commutation. (English) Zbl 1078.46044

Summary: 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 C * -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 A and B extends to an uncoupled product state on D; (ii) there is a representation π of D such that π(A) and π(B) commute and π is faithful on both A and B; (iii) A 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.

MSC:
46L30States of C * -algebras
46L60Applications of selfadjoint operator algebras to physics
81R15Operator algebra methods (quantum theory)