an:02133172
Zbl 1078.46044
Bunce, L. J.; Hamhalter, J.
\(C^*\)-independence, product states and commutation
EN
Ann. Henri PoincarĂ© 5, No. 6, 1081-1095 (2004).
1424-0637 1424-0661
2004
j
46L30 46L60 81R15
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 [\textit{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.
0197.26303