zbMATH — the first resource for mathematics

\(C^*\)-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 \(\pi\) 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.

46L30 States of selfadjoint operator algebras
46L60 Applications of selfadjoint operator algebras to physics
81R15 Operator algebra methods applied to problems in quantum theory
PDF BibTeX Cite
Full Text: DOI