A contribution to Segal algebras

Let A and B be Banach algebras. If B is an abstract Segal algebra in A, we have a bijective correspondence between the strictly irreducible representations of A and those of B. This gives a bijective correspondence for maximal modular left ideals. If A and B have approximate right units, we obtain a bijective correspondence for right resp. Two-sided ideals. For two-sided ideals this correspondence preserves the property of an ideal having approximate right units. This generalizes a Theorem by H. Reiter.It would be sufficient to require B to be a Banachspace.