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

DOI:
https://doi.org/10.1090/S0002-9939-1989-0973842-7

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 [1]). It is possible to define different types of unbounded commutants of an -algebra . We follow the definitions given in [2]. 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 [2]). 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 [2]).

**[1]**Robert T. Powers,*Self-adjoint algebras of unbounded operators*, Comm. Math. Phys.**21**(1971), 85–124. MR**0283580****[2]**Konrad Schmüdgen,*Strongly commuting selfadjoint operators and commutants of unbounded operator algebras*, Proc. Amer. Math. Soc.**102**(1988), no. 2, 365–372. MR**921001**, https://doi.org/10.1090/S0002-9939-1988-0921001-5

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DOI:
https://doi.org/10.1090/S0002-9939-1989-0973842-7

Article copyright:
© Copyright 1989
American Mathematical Society