On dense ideals of $C^\ast $-algebras and generalizations of the Gelfand–Naimark Theorem
We develop the theory of Segal algebras of commutative $C^\ast $-algebras, with an emphasis on the functional representation. Our main results extend the Gelfand–Naimark Theorem. As an application, we describe faithful principal ideals of $C^\ast $-algebras. A key ingredient in our approach is the use of Nachbin algebras to generalize the Gelfand representation theory.