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Transactions of the American Mathematical Society

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

 

 

A property for inverses in a partially ordered linear algebra


Authors: Taen Yu Dai and Ralph DeMarr
Journal: Trans. Amer. Math. Soc. 215 (1976), 285-292
MSC: Primary 06A70
DOI: https://doi.org/10.1090/S0002-9947-1976-0382116-2
MathSciNet review: 0382116
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Abstract: We consider a Dedekind $ \sigma $-complete partially ordered linear algebra A which has the following property: if $ x \in A$ and $ 1 \leqslant x$, then $ - u \leqslant {x^{ - 1}}$, where $ u = {u^2}$. This property is used to show that A must be commutative. We also show that A is the direct sum of two algebras, each of which behaves like an algebra of real-valued functions.


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DOI: https://doi.org/10.1090/S0002-9947-1976-0382116-2
Keywords: Dedekind $ \sigma $-complete partially ordered linear algebra, algebra of real-valued functions, f-ring, matrix inequalities
Article copyright: © Copyright 1976 American Mathematical Society