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Topologically nilpotent Banach algebras and factorisation

Published online by Cambridge University Press:  14 November 2011

P. G. Dixon
P. G. Dixon
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, England, U.K.
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Synopsis

A Banach algebra A is said to be topologically nilpotent if sup {‖x1x2…xn‖1/n: xi ∈ A, ‖xi‖ ≦ 1 (1 ≦ i ≦ n)} tends to zero as n → ∞. A Banach algebra A is uniformly topologically nil if sup {‖xn‖ 1/n: x ∈ A, ‖x‖ ≦ 1} tends to zero as n → ∞. These notions are equivalent for commutative algebras and a topological version of the Nagata-Higman Theorem gives a partial result for the non-commutative case. Topologically nilpotent algebras have a strong non-factorisation property and this yields theorems of the type “factorisation implies the existence of arbitrarily slowly decreasing powers”. Extensions of topologically nilpotent algebras by topologically nilpotent algebras are topologically nilpotent.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 3-4 , 1991 , pp. 329 - 341
Copyright
Copyright © Royal Society of Edinburgh 1991

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