Abstract.
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 [11] 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) \(A \otimes _{{\text{min}}} B\) 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.
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Communicated by Klaus Fredenhagen
submitted 03/12/03, accepted 26/04/04
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Bunce, L.J., Hamhalter, J. C*-Independence, Product States and Commutation. Ann. Henri Poincaré 5, 1081–1095 (2004). https://doi.org/10.1007/s00023-004-0191-7
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DOI: https://doi.org/10.1007/s00023-004-0191-7