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A non-commutative analog of the theorem concerning the decomposition of a measure. (Russian) Zbl 0571.46041

The paper gives an improved version, as well as detailed proofs, of some results contained in A. A. Lodkin, Funkts. Anal. Prilozh. Nov. 8, 54-58 (1974; Zbl 0312.46080). It is shown that for a finite measure on the lattice of projections of a von Neumann algebra \({\mathcal A}\) and any countably generated uniformly closed subalgebra of \({\mathcal A}\), there exists a system of conditional measures defined in an essentially unique way. Moreover, if the algebra \({\mathcal A}\) acts in a separable Hilbert space and almost all algebras \({\mathcal A}(\lambda)\) in the direct integral decomposition \({\mathcal A}=\int^{\oplus}_{\Lambda}{\mathcal A}(\lambda)dm(\lambda)\) are of type I, then such a system exists for the algebra \({\mathcal A}\) itself.
Reviewer: A.Luczak

MSC:

46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46L45 Decomposition theory for \(C^*\)-algebras

Citations:

Zbl 0312.46080
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