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Transactions of the American Mathematical Society

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

 

 

Inverse producing extension of a Banach algebra which eliminates the residual spectrum of one element


Author: C. J. Read
Journal: Trans. Amer. Math. Soc. 286 (1984), 715-725
MSC: Primary 46J05
DOI: https://doi.org/10.1090/S0002-9947-1984-0760982-0
MathSciNet review: 760982
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Abstract: If $ A$ is a commutative unital Banach algebra and $ G \subset A$ is a collection of nontopological zero divisors, the question arises whether we can find an extension $ A\prime$ of $ A$ in which every element of $ G$ has an inverse. Shilov [1] proved that this was the case if $ G$ consisted of a single element, and Arens [2] conjectures that it might be true for any set $ G$. In [3], Bollobás proved that this is not the case, and gave an example of an uncountable set $ G$ for which no extension $ A\prime$ can contain inverses for more than countably many elements of $ G$. Bollobás proved that it was possible to find inverses for any countable $ G$, and gave best possible bounds for the norms of the inverses in [4].

In this paper, it is proved that inverses can always be found if the elements of $ G$ differ only by multiples of the unit; that is, we can eliminate the residual spectrum of one element of $ A$. This answers the question posed by Bollobás in [5].


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DOI: https://doi.org/10.1090/S0002-9947-1984-0760982-0
Article copyright: © Copyright 1984 American Mathematical Society