\(C^*\)-independence, product states and commutation.

*(English)*Zbl 1078.46044Summary: 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 \(\pi\) of \(D\) such that \(\pi(A)\) and \(\pi(B)\) commute and \(\pi\) is faithful on both \(A\) and \(B\); (iii) \(A \otimes_{\min} B\) 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.

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.