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Une relation de chaine pour les dérivées de Radon-Nikodym spatiales. (A chain rule for spatial Radon-Nikodym derivatives). (French) Zbl 0635.46056
A chain rule for A. Connes’ spatial Radon-Nikodym derivative is obtained and an explicit formula for the canonical isomorphism of two Connes- Hilsum L p-spaces associated with the same von Neumann algebra is given. To this aim, the theory of relative tensor product of Hilbert spaces, developed by the author in the paper J. Operator Theory 9, 237-252 (1983; Zbl 0517.46050) is used. Due to a factorization property of relative tensor products, the general chain rule may be applied to find a relation between \(d\phi /d\psi_ 1\) and \(d\phi /d\psi_ 2\), where \(\phi\) is a normal semi finite weight on a von Neumann algebra M represented in two Hilbert spaces \({\mathcal H}_ 1\), \({\mathcal H}_ 2\), with \(\psi_ 1\), \(\psi_ 2\) being n.s.f. weights on the commutants \({\mathcal L}_ M(H_ 1)\), \({\mathcal L}_ M(H_ 2)\) of M in the spaces, respectively. The relation, in turn, allows to write the canonical isomorphism of L \(p(\psi_ 1)\) and L \(p(\psi_ 2)\), which takes \(u(d\phi /d\psi_ 1)^{1/p}\) into \(u(d\phi /d\psi_ 2)^{1/p}\) (u\(\in M\), \(\phi \in M_*\) \(+\), u \(*u=\sup p \phi)\), in the form of a relative ampliation.
Reviewer: St.Goldstein

MSC:
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46M05 Tensor products in functional analysis
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[1] CONNES (A.) , On the spatial theory of von Neumann algebras , J. F. A., vol. 35/2, 1980 , p. 153-164. MR 81g:46083 | Zbl 0443.46042 · Zbl 0443.46042 · doi:10.1016/0022-1236(80)90002-6
[2] CONNES (A.) . - Notes manuscrites , 1980 .
[3] HAAGERUP U. . - Lp-spaces associated with a von Neumann algebra , Preprint, Odense Universiteit, Denmark. · Zbl 0426.46045
[4] HILSUM (M.) . - Les espaces Lp d’une algèbre de von Neumann , J. F. A., vol. 40/2, 1981 , p. 151-169. MR 83c:46053 | Zbl 0463.46050 · Zbl 0463.46050 · doi:10.1016/0022-1236(81)90065-3
[5] SAUVAGEOT (J.-L.) . - Sur le produit tensoriel relatif d’espaces de Hilbert , J.O.T., vol. 9, 1983 , p. 237-252. MR 85a:46034 | Zbl 0517.46050 · Zbl 0517.46050
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