We give a generalization of the Stone-Weierstrass property for subalgebras of C(X), with X a completely regular Hausdorff space. In particular, we study in this paper some subalgebras of C0(X), with X a locally compact Hausdorff space, provided with weighted norm topology. By using the Stone-Weierstrass property, we then describe the ideal structure of these algebras.

We develop the theory of Segal algebras of commutative C * -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 * -algebras.A key ingredient in our approach is the use of Nachbin algebras to generalize the Gelfand representation theory.2010 Mathematics Subject Classification: 46J10, 46J25, 46J30.

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Abstract. We study subalgebras of C b (X) equipped with topologies that generalize both the uniform and the strict topology. In particular, we study the Stone-Weierstrass property and describe the ideal structure of these algebras.1. Introduction. Let X be a completely regular Hausdorff space. The algebra C b (X) of all continuous and bounded complex-valued functions on X is one of the most studied objects in modern analysis. Usually it is equipped with the supremum norm topology (denoted by σ) which makes it a Banach algebra. However, the structure of (C b (X), σ) is quite complicated. For example, its Gelfand space (the set of all regular maximal ideals equipped with the relative weak * -topology) is homeomorphic to the Stone-Čech compactification β(X) of X. Another difficulty is that (C b (X), σ) does not have the Stone-Weierstrass property, i.e., a point-separating symmetric subalgebra which is bounded away from zero is not necessarily uniformly dense in C b (X). So the topology on C b (X) defined by the supremum norm is not the "best" topology from this point of view.Another well-known and useful topology on C b (X) is the so-called strict topology (denoted by β) defined by the family of weighted supremum seminorms with weights running through all bounded (or equivalently, upper semicontinuous) non-negative functions on X which vanish at infinity. The topological algebra (C b (X), β) is in some sense easier to handle than (C b (X), σ). For example, its Gelfand space is homeomorphic to X and it has the Stone-Weierstrass property. On the other hand, even though the structure of (C b (X), β) has been extensively studied, it appears that its closed ideals have not been described yet. In [23] it was claimed without proof that in the case when X is locally compact, every closed ideal of (C b (X), β) consists of those functions on C b (X) which vanish on some closed subset E of X.