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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On $ \mathcal{A}$-submodules for reflexive operator algebras


Author: De Guang Han
Journal: Proc. Amer. Math. Soc. 104 (1988), 1067-1070
MSC: Primary 47D25
DOI: https://doi.org/10.1090/S0002-9939-1988-0969048-7
MathSciNet review: 969048
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Abstract: In [2] the authors described all weakly closed $ \mathcal{A}$-submodules of $ L\left( H \right)$ for a nest algebra $ \mathcal{A}$ in terms of order homomorphisms of Lat $ \mathcal{A}$. In this paper we prove that for any reflexive algebra $ \mathcal{A}$ which is $ \sigma $-weakly generated by rank-one operators in $ \mathcal{A}$, every $ \sigma $-weakly closed $ \mathcal{A}$-submodule can be characterized by an order homomorphism of Lat $ \mathcal{A}$. In the case when $ \mathcal{A}$ is a reflexive algebra with a completely distributive subspace lattice and $ \mathcal{M}$ is a $ \sigma $-weakly closed ideal of $ \mathcal{A}$, we obtain necessary and sufficient conditions for the commutant of $ \mathcal{A}$ modulo $ \mathcal{M}$ to be equal to AlgLat $ \mathcal{M}$.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0969048-7
Keywords: Reflexivity, submodule, commutant
Article copyright: © Copyright 1988 American Mathematical Society