# 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.

## Abstract

A Banach algebra *A* is said to be **topologically nilpotent** if

sup{||*x*_1*x*_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.

## Availability

Reprints of the paper available on request: e-mail P.Dixon with extension
@sheffield.ac.uk.

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