A note on topologically nilpotent Banach algebras

by P.G.Dixon and V. Müller.

Studia Math., 102 (1992), 269-275.

1991 Mathematics Subject Classification: 46H05


A Banach algebra A is said to be topologically nilpotent if

sup{||x_1x_2... x_n||^{1/n}: x_i \in A, ||x_i|| < 1 (1 < i < n)}

tends to zero as n \to \infty.

A Banach algebra A is uniformly topologically nil if

sup{||x^n||^{1/n}: x \in A,\; ||x|| < 1}

tends to zero as n \to \infty.

This paper continues the earlier study of these algebras. In particular, an example is given of a (necessarily non-commutative) Banach algebra which is uniformly topologically nil but not topologically nilpotent.


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