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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A commutative Banach algebra which factorizes but has no approximate units
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by Michael Leinert PDF
Proc. Amer. Math. Soc. 55 (1976), 345-346 Request permission

Abstract:

It is well known that any Banach algebra having bounded approximate units factorizes. For some time it was not clear if, conversely, factorization implied the existence of bounded approximate units. This was disproved by Paschke [3], but the problem remained open for commutative Banach algebras. We give an example of a commutative semisimple Banach algebra which factorizes but has not even unbounded approximate units.
References
  • Paul J. Cohen, Factorization in group algebras, Duke Math. J. 26 (1959), 199–205. MR 104982
  • Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
  • William L. Paschke, A factorable Banach algebra without bounded approximate unit, Pacific J. Math. 46 (1973), 249–251. MR 324413
Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 55 (1976), 345-346
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0397312-3
  • MathSciNet review: 0397312