A non-commutative Banach algebra in which 0 is the only quasi-nilpotent.

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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 B be the algebra of all infinite series x = ∑_{n=1}^{∞} α_{n}w_{n}, where ||x|| = ∑_{n=1}^{∞} ∥α_{n}| < ∞. Then B is a non-commutative Banach algebra. Let x ∈ B, x ≠ 0, and let α_{p} be the first non-zero coefficient in the series ∑_{n=1}^{∞} α_{n}w_{n}. Then the coefficient of w_{p}^{m} in x^{m} is precisely α_{p}^{m} and so ||x^{m}|| ≥ |α_{p}|^{m} (m = 1, 2, 3, …), r(x) ≥ |α_{p}| > 0. Note that B is an infinite dimensional non-commutative Banach algebra in which the set of quasi-nilpotents coincides with the set of nilpotents. 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.