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On the statistical independence of algebras of observables. (English) Zbl 0899.46054
The authors obtain a number of interesting results on independence in the context of C * - and W * -algebras, complementing and generalizing the results of the second-named author in Rev. Math. Phys. 2, 201-247 (1990; Zbl 0743.46079). It is shown that, for C * -subalgebras A, B of a C * -algebra C, their C * -independence is equivalent to the multiplicativity of the norm: for all aA and bB, ab=ab, even if the subalgebras do not commute. For a pair of commuting von Neumann subalgebras of a von Neumann algebra, the following notions of independence coincide: C * -independence, W * -independence, strict locality in both C * - and W * -algebra sense, Schlieder’s condition and the existence of a product extension. Some counterexamples are given and the question of kinematical independence is discussed. The results are further complemented by those of J. Hamhalter in Ann. Inst. Henri Poincaré, Phys. Theor. 67, No. 4, 447-462 (1997).
46L60Applications of selfadjoint operator algebras to physics
46L10General theory of von Neumann algebras
81T05Axiomatic quantum field theory; operator algebras