A Banach algebra having no bounded approximate identity.

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{xy ; x, y ∈ l^{2}} is a proper subset of Banach algebra l^{2} equipped with the coordinatewise operations. In fact (\frac{1}{n}) ∈ l^{2} and if x_{n}y_{n} = \frac{1}{n}, then there exist an integer N such that for all n > N, |x_{n}| ≥ \frac{1}{\sqrt{n}} or for all n > N, |y_{n}| ≥ \frac{1}{\sqrt{n}}, and hence (x_{n}) ∉ l^{2} or (y_{n}) ∉ l^{2}. Now Cohen’s factorization theorem [B&D,§11. Corollary 11] implies that l^{2} has no bounded approximate identity.

Comment. Using BA37, we conclude that the Banach algebra l^{2} has neither bounded aproximate identity nor unbounded one. Ref. [B&D] F.F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, 1973.