Adjoining inverses to commutative Banach algebras

Author:
Béla Bollobás

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

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Abstract: Let be a commutative unital Banach algebra. Suppose is such that for all . Two questions are considered in the paper. Does there exist a superalgebra of in which every is invertible? Can one always have also if ? Arens proved that if then there is an algebra containing , with . In the paper it is shown that if is countable exists, but if is uncountable, this is not necessarily so. The answer to the second question is negative even if consists of only two elements.

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DOI:
https://doi.org/10.1090/S0002-9947-1973-0324418-9

Article copyright:
© Copyright 1973
American Mathematical Society