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Zbl 0899.46054
Florig, Martin; Summers, Stephen J.
On the statistical independence of algebras of observables. (English)
[J] J. Math. Phys. 38, No.3, 1318-1328 (1997). ISSN 0022-2488; ISSN 1089-7658/e

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 $a\in A$ and $b\in B$, $\| ab\| =\| a\| \| b\| $, 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 {\it J. Hamhalter} in Ann. Inst. Henri Poincaré, Phys. Theor. 67, No. 4, 447-462 (1997).
[S.Goldstein (Łódź)]

MSC 2000:: *46L60 Appl. of selfadjoint operator algebras to physics
46L10 General theory of von Neumann algebras
81T05 Axiomatic quantum field theory

Keywords: von Neumann algebra; $C^*$-algebra; $C^*$-independence; statistical independence; Schlieder property; $W^*$-independence; kinematical independence; multiplicativity; strict locality

Citations: Zbl 0743.46079

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