On the strong unbounded commutant of an -algebra
Authors: A. Van Daele and A. Kasparek
Journal: Proc. Amer. Math. Soc. 105 (1989), 111-116
MSC: Primary 47D40; Secondary 46K10
MathSciNet review: 973842
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Abstract: Let be a dense subspace of a Hilbert space . An -algebra on is a -algebra of linear operators defined on and leaving invariant which contains the identity map of . The involution in is the map (see ). It is possible to define different types of unbounded commutants of an -algebra . We follow the definitions given in . Let be the vector space of all continuous linear mappings of into with respect to the graph topology on introduced by the operators from . Then the strong unbounded commutant is defined as for all . is an algebra, but in general however it will not be -invariant (see ). In this paper we show that even worse can happen. We give an example of such an -algebra and an operator such that .
In particular this shows that the strong unbounded commutant of an -algebra may contain operators which are not closable. Furthermore the constructed -algebra gives another example for an -algebra whose socalled form commutant contains a sesquilinear form which is not an operator (see ).
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