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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The algebra of decomposable operators in direct integrals of not necessarily separable Hilbert spaces


Author: Reinhard Schaflitzel
Journal: Proc. Amer. Math. Soc. 110 (1990), 983-987
MSC: Primary 47D25; Secondary 04A30, 46L45, 47B40
MathSciNet review: 1028294
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Abstract: Considering direct integrals of not necessarily separable Hilbert spaces we examine the question whether the algebra of decomposable operators is the commutant of the algebra of diagonalizable operators. Using the continuum-hypothesis we prove this relation, if the set of square integrable vector fields is generated by a subset $ {\Gamma _0}$ such that $ \vert{\Gamma _0}\vert \leq \vert{\mathbf{R}}\vert$. For the general case, a counterexample is given.


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DOI: https://doi.org/10.1090/S0002-9939-1990-1028294-6
Keywords: Direct integrals, algebra of decomposable operators, algebra of diagonalizable operators, continuum-hypothesis
Article copyright: © Copyright 1990 American Mathematical Society