Topologically nilpotent Banach algebras and factorization

by P.G.Dixon

Proc. Roy. Soc. Edin. A, 119 (1991), 329-341.

1985 Mathematics Subject Classification: primary 46H05; secondary 46H25, 16A22.


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.

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-factorization property and this yields theorems of the type ``factorization implies the existence of arbitrarily slowly decreasing powers''. Extensions of topologically nilpotent algebras by topologically nilpotent algebras are topologically nilpotent.


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