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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.
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