Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 0324418
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46J05

Retrieve articles in all journals with MSC: 46J05

Additional Information

Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society