Vol. 61, No. 2, 1975

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A Radon Nikodým theorem for weights on von Neumann algebras

Alfons Van Daele

Vol. 61 (1975), No. 2, 527–542

Let φ and ψ be normal positive linear functionals on a von Neumann algebra M such that ψ φ. Sakai proved the existence of a unique element h M with 0 h 1 such that ψ(x) = 1
2φ(hx + xh) for any x M. A generalization of this theorem is obtained for weights on von Neumann algebras. Let φ be a faithful normal semi-finite weight and ψ any weight on M majorized by φ. Then there is a unique element h M with 0 h 1 such that ψ(x) = 1
2φ(hx + xh) holds for x in a σ-weakly dense *-subalgebra of M. A stronger version is obtained when ψ is assumed to be a normal positive linear functional. Moreover counterexamples are given to show that in general one can not expect this relation to hold for every x M+.

Mathematical Subject Classification 2000
Primary: 46L10
Received: 13 May 1974
Published: 1 December 1975
Alfons Van Daele
Department of Mathematics
Katholieke Universiteit Leuven
3030 Leuven