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*_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.

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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