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Classes of Invariant Subspaces for Some Operator Algebras

International Journal of Theoretical Physics volume 53, pages 3397–3408 (2014)Cite this article

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Abstract

New results showing connections between structural properties of von Neumann algebras and order theoretic properties of structures of invariant subspaces given by them are proved. We show that for any properly infinite von Neumann algebra M there is an affiliated subspace \({\mathcal{L}} \) such that all important subspace classes living on \({\mathcal{L}} \) are different. Moreover, we show that \({\mathcal{L}} \) can be chosen such that the set of σ-additive measures on subspace classes of \({\mathcal{L}} \) are empty. We generalize measure theoretic criterion on completeness of inner product spaces to affiliated subspaces corresponding to Type I factor with finite dimensional commutant. We summarize hitherto known results in this area, discuss their importance for mathematical foundations of quantum theory, and outline perspectives of further research.

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Acknowledgements

This work was supported by the ’Grant Agency of the Czech Republic’ grant number P201/12/0290, “Topological and geometrical properties of Banach spaces and operator algebras”.

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Authors and Affiliations

  1. Faculty of Electrical Engineering, Department of Mathematics, Czech Technical University, Technicka 2, 166 27, Prague 6, Czech Republic

    Jan Hamhalter

  2. Institute of Computer Mathematics and Information Technologies, Kazan Federal University, Kremlevskaya 18, Kazan, Russia

    Ekaterina Turilova

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  1. Jan Hamhalter
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  2. Ekaterina Turilova
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Correspondence to Ekaterina Turilova.

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Hamhalter, J., Turilova, E. Classes of Invariant Subspaces for Some Operator Algebras. Int J Theor Phys 53, 3397–3408 (2014). https://doi.org/10.1007/s10773-013-1740-y

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  • DOI: https://doi.org/10.1007/s10773-013-1740-y

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