Let M be a von Neumann algebra with a von Neumann subalgebra MQ. If £ is a conditional expectation (i.e., projection of norm one) from M into Λ/ o , then any faithful normal state φ 0 admits a natural extension φo o E with respect to E in the sense that E = E φQ. E. If E ω is only an ω-conditional expectation, then φ 0 o E ω is not always an extension of φ 0. This paper is devoted to the construction of an extension φo of ψo generalizing the above situation for ω-conditional expectations, which leads also to a Radon-Nikodym theorem for ωconditional expectation under suitable majorization condition. 10 CARLO CECCHINI AND DENES PETZ E ω = E ψ then (φo)~ω = {φo)~ψ In general, E ω (v*aυ) = E ψ {ά) where v is an appropriate isometry in M and ψ stands for (φ o)~ω. Our references on von Neumann algebras and their modular theory are [14] and [15]. We use the standard notations of the Tomita-Takesaki theory without any explanation. H will denote always a Hubert space and if M c B(H) then M 1 is the commutant of M. For the sake of convenience, states on M 1 are marked with a prime, for example ω 1 etc. The main results are contained in § §3 and 4.
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