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In this note we obtain an elementary extension of the so-called Fundamental Theorem of Segal Algebras to a special class of noncommutative Banach algebras. In particular the case of noncommutative group algebras is settled.

In this note we unify and simplify some recent results showing the impossibility of factoring in certain convolution subalgebras of the group algebra of a nondiscrete LCAG. A new result is a direct proof of nonfactorization of the classical Hardy spaces, regarded as convolution algebras, on the circle. By considering the ideal of Hilbert-Schmidt operators in the algebra of compact operators on a Hilbert space we illustrate that nonfactorization is not peculiar to convolution.