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, u^{2}, uv, v^{2}, u^{3}, u^{2}v, . . .. Let γ(w_{n}) denote the length of the word w_{n}, and let C be the algebra of all infinite series x = ∑_{n=1}^{∞} α_{n}w_{n} where ∥x_{n}∥ = ∑ \frac{|α_{n}|}{γ(w_{n})!} < ∞. Then C is clearly a non-commutative Banach algebra and an integral domain. Let x ∈ C and let k be a positive integer. We have

∥x^{k}∥ ≤ ∑_{n_{i}} \frac{|α_{n_{1}}||α_{n_{2}}| … |α_{n_{k}}|}{γ(w_{n_{1}}w_{n_{2}} … w_{n_{k}})!}
= ∑_{n_{i}} \frac{γ(w_{n_{1}})! … γ(w_{n_{k}})!|α_{n_{1}}|}{\{γ(w_{n_{1}}) + … + γ(w_{n_{k}})\}!γ(w_{n_{1}})!} … \frac{|α_{n_{k}}|}{γ(w_{n_{k}})!}
≤ \frac{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.