Ideals in CB(X) arising from ideals in X

Abstract

Let X be a completely regular topological space. We assign to each (set theoretic) ideal of X an (algebraic) ideal of C B (X), the normed algebra of continuous bounded complex valued mappings on X equipped with the supremum norm. We then prove several representation theorems for the assigned ideals of C B (X). This is done by associating a certain subspace of the Stone-Čech compactification βX of X to each ideal of X. This subspace of βX has a simple representation, and in the case when the assigned ideal of C B (X) is closed, coincides with its spectrum as a C * -subalgebra of C B (X). This in particular provides information about the spectrum of those closed ideals of C B (X) which have such representations. This includes the nonvanishing closed ideals of C B (X) whose spectrums are studied in great detail. Our representation theorems help to understand the structure of certain ideals of C B (X). This has been illustrated by means of various examples. Our approach throughout will be quite topological and makes use of the theory of the Stone-Čech compactification.

On closed non-vanishing ideals in C(X) II; compactness properties

“…In [7], the second author has studied closed non-vanishing ideals of C B (X), where X is a completely regular space, by relating them to certain subspaces of the Stone-Čech compactification of X. The precise statement is as follows.…”

“…The simple structure of sp(H) in the above theorem enables us to study its properties. This is done by relating (topological) properties of sp(H) to (algebraic) properties of H. Among various results of this type in [7], the following two theorems are concerned with connectedness properties of sp(H). Theorem 2.2.…”

“…The above few theorems serve as a starting point for our present study of nonvanishing closed ideals of C B (X). Examples of non-vanishing closed ideals in C B (X) (for various completely regular spaces X) may be found in Part 3 of [7]. (See [2], and [4]- [6] for other relevant examples and results.…”

“…Here, as usual, X is a completely regular space and C B (X) is endowed with pointwise addition and multiplication and the supremum norm. Our theorem is motivated by Theorem 3.2.12 of [7] (Theorem 2.3) which gives a necessary and sufficient condition such that all (connected) components of the spectrum of a non-vanishing closed ideal of C B (X) are open. (Observe that for a space Y the requirement that all components are open is equivalent to saying that every point of Y has a connected open neighborhood.…”

“…This is done by studying the (unique) locally compact Hausdorff space Y which is associated to H in such a way that H and C 0 (Y ) are isometrically isomorphic. The space Y is constructed by the second author in [7] as a subspace of the Stone-Čech compactification of X, and coincides with the spectrum of H (considered as a Banach algebra) in the case when the field of scalars is C. The known (and simple) structure of Y enables us to study its properties. We are particularly interested in connectedness properties of Y .…”