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{xy ; x,y Î l2} is a proper subset of Banach algebra l2 equipped
with the coordinatewise operations. In fact ([ 1/ n]) Î l2 and if xnyn = [ 1/ n], then there exist an integer N such that for all
n > N, |xn| ³ [ 1/( Ön)] or for all
n > N, |yn| ³ [ 1/( Ön)], and hence
(xn) Ï l2 or (yn) Ï l2. Now Cohen's factorization theorem
[B&D,§11. Corollary 11] implies that l2 has no bounded approximate
identity.
Comment. Using BA37, we conclude that the Banach algebra l2 has neither bounded aproximate identity nor unbounded one.
Ref.
[B&D] F.F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, 1973.