A Banach algebra A in which every singular element is a left or right topological divisor of zero.

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Let A = B(X), the Banach algebra of bounded linear mappings from a Banach space X into X, and T ∈ Sing(A). For y ∈ X and g ∈ X, the rank one operator y\overline{⊗}g ∈ B(X) is defined by (y\overline{⊗}g)(z) = g(z)y (z ∈ X). If T isn’t 1-1, then there exists an element x ≠ 0 such that T x = 0. So if f ∈ X^{#} and f(x) = 1, then T (x\overline{⊗}f) = T x\overline{⊗}f = 0. So T is a left divisor of zero. If T (X) ≠ X and T (X)^{−} ≠ X, then by the Hahn-Banach theorem there exists a non-zero functional f such that f(T (X)) = 0. Therefore (x\overline{⊗}f)T = 0, for all x ∈ X. Thus T is a right divisor of zero.

Finally if T (X) ≠ X and T (X)^{−} = X, then there exists a sequence (y_{n}) in X satisfying ||y_{n}|| = 1 and T y_{n} ⟶ 0. If f ∈ X^{#} with ||f|| = 1 and U_{n} = y_{n}\overline{⊗}f, then ||U_{n}|| = ||y_{n}|| ||f|| = 1 and ||T U_{n}|| = ||T y_{n}\overline{⊗}f|| ≤ ||T y_{n}|| and hence T U_{n} ⟶ 0. Thus T is a right topological divisor of zero.