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Proceedings of the American Mathematical Society

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A commutative Banach algebra which factorizes but has no approximate units

Author: Michael Leinert
Journal: Proc. Amer. Math. Soc. 55 (1976), 345-346
MathSciNet review: 0397312
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Abstract | References | Additional Information

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

  • [1] P. J. Cohen, Factorization in group algebras, Duke Math. J. 26(1959), 199-205. MR21 #3729. MR 0104982 (21:3729)
  • [2] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der math. Wissenschaften, Band 152, Springer-Verlag, Berlin and New York, 1970. MR41 #7378; erratum, 42, p. 1825. MR 0262773 (41:7378)
  • [3] W. L. Paschke, A factorable Banach algebra without bounded approximate unit, Pacific J. Math. 46(1973), 249-251. MR48 #2765. MR 0324413 (48:2765)

Additional Information

Keywords: Banach algebra, factorization, approximate units
Article copyright: © Copyright 1976 American Mathematical Society

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