The algebra of decomposable operators in direct integrals of not necessarily separable Hilbert spaces
HTML articles powered by AMS MathViewer
- by Reinhard Schaflitzel PDF
- Proc. Amer. Math. Soc. 110 (1990), 983-987 Request permission
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 $|{\Gamma _0}| \leq |{\mathbf {R}}|$. For the general case, a counterexample is given.References
- Jacques Dixmier, von Neumann algebras, North-Holland Mathematical Library, vol. 27, North-Holland Publishing Co., Amsterdam-New York, 1981. With a preface by E. C. Lance; Translated from the second French edition by F. Jellett. MR 641217
- Esben T. Kehlet, Disintegration theory on a constant field of nonseparable Hilbert spaces, Math. Scand. 43 (1978), no. 2, 353–362 (1979). MR 531315, DOI 10.7146/math.scand.a-11789
- Odile Maréchal, Champs mesurables d’espaces hilbertiens, Bull. Sci. Math. (2) 93 (1969), 113–143 (French). MR 261333
- Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728
- Jørgen Vesterstrøm and Wilbert Wils, Direct integrals of Hilbert spaces. II, Math. Scand. 26 (1970), 89–102. MR 264416, DOI 10.7146/math.scand.a-10968
- Wilbert Wils, Direct integrals of Hilbert spaces. I, Math. Scand. 26 (1970), 73–88. MR 264415, DOI 10.7146/math.scand.a-10967
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 983-987
- MSC: Primary 47D25; Secondary 04A30, 46L45, 47B40
- DOI: https://doi.org/10.1090/S0002-9939-1990-1028294-6
- MathSciNet review: 1028294