Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-07T07:14:14.770Z Has data issue: false hasContentIssue false
  • English
  • Français

Generators of reflexive algebras

Published online by Cambridge University Press:  09 April 2009

W. E. Longstaff
W. E. Longstaff
Affiliation:
The Flinders University of South Australia Bedford Park, 5042, South Australia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For any collection of closed subspaces of a complex Hilbert space the set of bounded operators that leave invariant all the members of the collection is a weakly-closed algebra. The class of such algebras is precisely the class of reflexive algebras as defined for example in Radjavi and Rosenthal (1969) and contains the class of von Neumann algebras.In this paper we consider the problem of when such algebras are finitely generated as weakly-closed algebras. It is to be hoped that analysis of this problem may shed some light on the famous unsolved problem of whether every von Neumann algebra on a separable Hilbert space is finitely generated. The case where the underlying space is separable and the collection of subspaces is totally ordered is dealt with in Longstaff (1974). In the present paper the result of Longstaff (1974) is generalized to the case of a direct product of countably many totally ordered collections each on a separable space. Also a method of obtaining non-finitely generated reflexive algebras is given.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 20 , Issue 2 , September 1975 , pp. 159 - 164
Copyright
Copyright © Australian Mathematical Society 1975

References

Dixmier, J. (1957), Les algèbres d'opérateurs dans l'espace Hilbertien (Gauthier-Villars, Paris, 1957.)Google Scholar
Longstaff, W. E. (1974), ‘Generators of nest algebras’, Canad. J. Math. 26, 565–575.CrossRefGoogle Scholar
Radjavi, H. and Rosenthal, P. (1969), ‘On invariant subspaces and reflexive algebras’, Amer. J. Math. 91, 683–692.CrossRefGoogle Scholar
Ringrose, J. R. (1965), ‘On some algebras of operators’, Proc. London Math. Soc. (3) 15, 61–83.CrossRefGoogle Scholar