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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Adjoining inverses to commutative Banach algebras
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by Béla Bollobás PDF
Trans. Amer. Math. Soc. 181 (1973), 165-174 Request permission

Abstract:

Let $A$ be a commutative unital Banach algebra. Suppose $G \subset A$ is such that $||a|| \leqslant ||ga||$ 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 $||{g^{ - 1}}|| \leqslant 1$ if $g \in G$? Arens proved that if $G = \{ g\}$ then there is an algebra containing ${g^{ - 1}}$, with $||{g^{ - 1}}|| \leqslant 1$. 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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Additional Information
  • © Copyright 1973 American Mathematical Society
  • 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