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

Read, C. J.

doi:10.1090/S0002-9947-1984-0760982-0

Inverse producing extension of a Banach algebra which eliminates the residual spectrum of one element
HTML articles powered by AMS MathViewer

by C. J. Read PDF

Trans. Amer. Math. Soc. 286 (1984), 715-725 Request permission

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].

References

On normed rings with one generator

21

Richard Arens, Linear topological division algebras, Bull. Amer. Math. Soc. 53 (1947), 623–630. MR 20987, DOI 10.1090/S0002-9904-1947-08857-1
Béla Bollobás, Adjoining inverses to commutative Banach algebras, Trans. Amer. Math. Soc. 181 (1973), 165–174. MR 324418, DOI 10.1090/S0002-9947-1973-0324418-9
Béla Bollobás, Best possible bounds of the norms of inverses adjoined to normed algebras, Studia Math. 51 (1974), 87–96. MR 348502, DOI 10.4064/sm-51-2-87-96

Adjoining inverses to commutative Banach algebras

John A. Lindberg Jr., Extension of algebra norms and applications, Studia Math. 40 (1971), 35–39. MR 313816, DOI 10.4064/sm-40-1-35-39