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Transactions of the American Mathematical Society

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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
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].

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

  • [1] G. E. Shilov, On normed rings with one generator, Mat. Sb. 21 (63) (1947), 25-46.
  • [2] Richard Arens, Linear topological division algebras, Bull. Amer. Math. Soc. 53 (1947), 623-630. MR 0020987 (9:6a)
  • [3] B. Bollobás, Adjoining inverses to Banach algebras, Trans. Amer. Math. Soc. 181 (1973), 165-174. MR 0324418 (48:2770)
  • [4] -, Best possible bounds to the norms of inverses adjoined to normed algebras, Studia Math. 51 (1974), 87-96. MR 0348502 (50:1000)
  • [5] -, Adjoining inverses to commutative Banach algebras, Algebras in Analysis (J. H. Williamson, ed.), Academic Press, New York, 1975, pp. 256-257.
  • [6] J. A. Lindberg, Extensions of algebra norms and applications, Studia Math. 40 (1971), 35-39. MR 0313816 (47:2370)

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