A non-commutative radical Banach algebra which

A non-commutative radical Banach algebra which A non-commutative radical Banach algebra which is an integral domain.

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Let A be the free algebra on two symbols w,v, i.e. the algebra of all finite linear combinations of words in u and v. The set of all such words is countable, {w_n}, and we take the standard enumeration given by u,v,u²,uv,v²,u³,u²v,¼. Let g(w_n) denote the length of the word w_n, and let C be the algebra of all infinite series x = å_{n = 1}^¥ a_n w_n where ||x_n|| = å[( |a_n|)/( g(w_n)!)] < ¥. Then C is clearly a non-commutative Banach algebra and an integral domain. Let x Î C and let k be a positive integer. We have

||x^k||

£

å
n_i
|a_n₁||a_n₂|¼|a_{n_k}|
g(w_n₁w_n₂¼w_{n_k})!

=

å
n_i
g(w_n₁)!¼g(w_{n_k})!|a_n₁|
{g(w_n₁)+¼+g(w_{n_k})}!g(w_n₁)!
¼ |a_{n_k}|
g(w_{n_k})!

£

1
k!
||x||^k.

Hence r(x) = 0.
Ref.
J. Duncan and A.W. Tullo, Finite dimensionality, nilpotents and quasi-nilpotents in Banach algebras, Proc. of the Edin. math. Soc., vol 19(Series II), Part 1, 1974.

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On 22 Feb 2001, 00:15.