Generalization of the topological algebra (C_b(X), β)

Abstract

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.

C∗-Segal Algebras with Order Unit, II

“…b (X) and C v 0 (X) are self-adjoint C*-Segal algebras. In fact, they are examples of the so-called Nachbin algebras; see, e.g., [17,2,3]. It is easy to see that the function 1 v serves as an order unit for C v b (X).…”

Description of Closed Ideals in Function Algebras Containing Continuous Unbounded Functions

“…In [9] proper subalgebras of C b (X) and their ideal structure were developed. In the present paper, the ideal structure of algebras of continuous functions on a completely regular Hausdorff space X containing unbounded functions (with respect to both topologies described above) are considered.…”

On dense ideals of C*-algebras and generalizations of the Gelfand–Naimark Theorem

“…In this section, we recall some basic facts about Nachbin algebras. For a more detailed discussion of these algebras, including full proofs, we refer to [4,5].…”