A commutative radical Banach algebra.

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1. A Banach space with all products taken to be zero. Then every element is quasi invertible.

2.The Banach space L^{1}([0, 1]) with the product (fg)(x) = ∫_{0}^{x} f(x − y)g(y)dy has f_{∘}(t) = t, 0 ≤ t ≤ 1 as a generator since f_{∘}^{n}(t) = \frac{t^{n−1}}{(n − 1)!}, the set of polynomials in one variable is L^{p}-dense in C([0, 1]) and C([0, 1]) is L^{p}-dense in L^{1}([0, 1]) (cf. [Rud2, Theorem 2.14]).

Moreover ∥ f_{∘}^{n} ∥= ∫_{0}^{1} |f_{∘}^{n}(t)|dt = \frac{1}{n!}, so r(f_{∘}) = lim_{n} ∥ f_{∘}^{n} ∥\frac{1}{n} = lim_{n}(n!)\frac{−1}{n} =

0. Therefore this algebra doesn’t have any character. Thus it is radical algebra.

Ref.

[Rud2] W. Rudin, Real and complex analysis, McGraw-Hill, 1986.