A non-topologically nilpotent Banach algebra.

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1. The algebra C of complex numbers.

2.The Volterra algebra L^{1}[0, 1]^{1} isn’t topologically nilpotent ; For establishing

this, consider x_{i}(t) = \{ \begin{matrix}2^{i} & 0 ≤ t ≤ 2^{−i} \\ 0 & 2^{−i} ≤ t ≤ 1 \end{matrix} .

Then ∥x_{i}∥ = 1 (i = 1, 2, …) and for all n, ∥x_{1} … x_{n}∥ = 1.

1The Banach space L^{1}([0, 1]) with the product (fg)(x) = ∫_{0}^{x} f(x − y)g(y)dy is a non-unital commutative Banach algebra and called Volterra algebra.