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Graded Banach algebras associated with varieties of Banach algebras

Published online by Cambridge University Press:  24 October 2003

P. G. DIXON
P. G. DIXON
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield, S3 7RH. e-mail: P.Dixon@sheffield.ac.uk
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Abstract

The concept of ‘varieties of Banach algebras’ was introduced by the author (Quart. J. Math. Oxford (2), 27 (1976), 481–487). Here we develop the theory, producing analogues of the ‘relatively free’ objects in varieties of universal algebras. We take as a test question the problem of describing all varieties which are closed under bicontinuous isomorphisms (i.e. varieties which are also semivarieties), and show that this may be answered easily with the aid of these new concepts. (The answer is, as expected, just those varieties which are ‘algebraically defined’.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 135 , Issue 3 , November 2003 , pp. 469 - 479
Copyright
© 2003 Cambridge Philosophical Society

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