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 is a commutative unital Banach algebra and is a collection of nontopological zero divisors, the question arises whether we can find an extension of in which every element of has an inverse. Shilov [**1**] proved that this was the case if consisted of a single element, and Arens [**2**] conjectures that it might be true for any set . In [**3**], Bollobás proved that this is not the case, and gave an example of an uncountable set for which no extension can contain inverses for more than countably many elements of . Bollobás proved that it was possible to find inverses for any countable , 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 differ only by multiples of the unit; that is, we can eliminate the residual spectrum of one element of . This answers the question posed by Bollobás in [**5**].

**[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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DOI:
https://doi.org/10.1090/S0002-9947-1984-0760982-0

Article copyright:
© Copyright 1984
American Mathematical Society