Vol. 39, No. 1, 1971

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An extension of some results of Takesaki in the reduction theory of von Neumann algebras

George A. Elliott

Vol. 39 (1971), No. 1, 145–148

Briefly, the results in this paper are that both for measurable fields of von Neumann algebras and for families of measurable fields of operators, pointwise isomorphism implies isomorphism.

In the special case when half the measurable fields considered are constant, these results were established by Takesaki. If the Borel space on which the fields are defined is standard, the results can be established by classical means; in the case considered by Takesaki they are due to von Neumann.

For the results of the present paper, two new tools seem to be needed. The first is a measurable choice theorem of Aumann which generalizes the classical one. This has already been applied to reduction theory by Flensted-Jensen. The second is a criterion for a von Neumann algebra containing the diagonal operators to be decomposable: it should consist of decomposable operators. This answers a question of Dixmier.

Mathematical Subject Classification 2000
Primary: 46L10
Received: 22 December 1970
Published: 1 October 1971
George A. Elliott