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Transactions of the American Mathematical Society

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

 

 

Adjoining inverses to commutative Banach algebras


Author: Béla Bollobás
Journal: Trans. Amer. Math. Soc. 181 (1973), 165-174
MSC: Primary 46J05
DOI: https://doi.org/10.1090/S0002-9947-1973-0324418-9
MathSciNet review: 0324418
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Abstract: Let $ A$ be a commutative unital Banach algebra. Suppose $ G \subset A$ is such that $ \vert\vert a\vert\vert \leqslant \vert\vert ga\vert\vert$ for all $ g \in G,a \in A$. Two questions are considered in the paper. Does there exist a superalgebra $ B$ of $ A$ in which every $ g \in G$ is invertible? Can one always have also $ \vert\vert{g^{ - 1}}\vert\vert \leqslant 1$ if $ g \in G$? Arens proved that if $ G = \{ g\} $ then there is an algebra containing $ {g^{ - 1}}$, with $ \vert\vert{g^{ - 1}}\vert\vert \leqslant 1$. In the paper it is shown that if $ G$ is countable $ B$ exists, but if $ G$ is uncountable, this is not necessarily so. The answer to the second question is negative even if $ G$ consists of only two elements.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0324418-9
Article copyright: © Copyright 1973 American Mathematical Society