A topologically nilpotent Banach algebra. (A A topologically nilpotent Banach algebra. (A Banach algebra A is called topologically nilpotent if the quantity NA(n) = sup{||x1...xn||[ 1/ n] ; xi Î A ;  ||xi|| £ 1 ,  1 £ i £ n} tends to zero as n® ¥).

******************************

The Banach algebra C[0,1] with the supremum norm ||.|| and convolution multiplication is topologically nilpotent:
Defining u Î C([0,1]) by u(t) = 1  (0 £ t £ 1), we have un(t) = [( tn-1)/( (n-1)!)] (n = 1,2,¼) and so ||un|| = [ 1/( (n-1)!)]. For arbitrary f1,¼,fn Î C([0,1]), |f1*f2*¼*fn)(t)| £ ||f1||¼||fn|| un(t). Hence ([( ||f1*¼*fn||)/( ||f1||¼||fn||)])[ 1/ n] £ [ 1/( ((n-1)!)[ 1/ n])]. Now note that limn [ 1/( ((n-1)!)[ 1/ n])] = 0.
Ref.
P.G. Dixon, G. A. Willis, Approximate identities in extensions of topological nilpotent Banach algebras., Proc. Royal of Edin., 122A, 45-52, 1992.

File translated from TEX by TTH, version 2.70.
On 22 Feb 2001, 00:15.