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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Complementation for right ideals in generalized Hilbert algebras


Author: John Phillips
Journal: Trans. Amer. Math. Soc. 197 (1974), 409-417
MSC: Primary 46K15
DOI: https://doi.org/10.1090/S0002-9947-1974-0385579-X
MathSciNet review: 0385579
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Abstract: Let $ \mathfrak{A}$ be a generalized Hilbert algebra and let $ \mathcal{J}$ be a closed right ideal of $ \mathfrak{A}$. Let $ {\mathcal{J}^ \bot }$ denote the pre-Hilbert space orthogonal complement of $ \mathcal{J}$ in $ \mathfrak{A}$. The problem investigated in this paper is: for which algebras $ \mathfrak{A}$ is it true that $ \mathfrak{A} = \mathcal{J} \oplus {\mathcal{J}^ \bot }$ for every closed right ideal $ \mathcal{J}$ of $ \mathfrak{A}$? In the case that $ \mathfrak{A}$ is achieved, a slightly stronger property is characterized and these characterizations are then used to investigate some interesting examples.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0385579-X
Article copyright: © Copyright 1974 American Mathematical Society