Local invertibility in subrings of C*(X)

Abstract

It is known that the maximal ideals in the rings C(X) and C*(X) of continuous and bounded continuous functions on X, respectively, are in one-to-one correspondence with j3X. We have shown previously that the same is true for any ring A(X) between C(X) and C*(X). Here we consider the problem for rings A(X) contained in C'{X) which are complete rings of functions (that is, they contain the constants, separate points and closed sets, and are uniformly closed). For every noninvertible / £ A(X), we define a z-filter Z>.A(/) on X which, in a sense, provides a measure of where / is 'locally invertible'. We show that the map ZA generates a correspondence between ideals of A(X) and z-filters on X. Using this correspondence, we construct a unique compactification of X for every complete ring of functions. Each such compactification is explicitly identified as a quotient of fiX. In fact, every compactification of X arises from some complete ring of functions A(X) via this construction. We also describe the intersections of the free ideals and of the free maximal ideals in complete rings of functions. INTRODUCTIONIf X is a completely regular space then the collection C(X) of continuous realvalued functions on X is a ring under pointwise operations. The ring C(X) and its subring C*(X), consisting of the bounded elements of C{X), have been studied extensively (see [4] and [1]). The resulting theory of 'rings of continuous functions' is beautifully presented in the now classic text of Gillman and Jerison by that title. One of the main achievements in this theory is the characterisation of the maximal ideals in the rings C(X) and C*(X). Although the problems of characterising the maximal ideals in C(X) and C*(X) are quite different, they have the same solution-namely, the set of maximal ideals in each is in one-to-one correspondence with /3X, the Stone-Cech compactification of X. In [6], a method was developed to characterise the maximal ideals in any ring A(X) of continuous functions on X with A(X) D C*(X), and in [2] the method was used to study deeper properties of these rings. The technique of 'local invertibility' used in these papers has the desirable effect of putting the problems of characterising the maximal ideals in C{X) and C*(X) into a common setting. This

Remarks on $LBI$-subalgebras of $C(X)$

“…Following [5] a subring A(X) of C * (X) is called a complete ring of functions if A(X) is a uniformly closed subset of C * (X), contains the constants and separates points and closed sets in X. Throughout this section A(X) denotes a subalgebra of C * (X) which is a complete ring of functions.…”

“…As every complete ring of functions is an LBI-subalgebra, the structure space of each complete ring of functions is a compactification of X and hence is a quotient of βX. This means that the compactification which is characterized in [5] via the mapping Z A for a complete ring of functions A(X) is, in fact, the structure space of A(X). Proposition 3.2.…”

“…The aim of this paper is to investigate a different approach to the results of [5] and [15]. This is done via characterizing maximal ideals of the subalgebras which are considered in the mentioned papers.…”

“…Using these, we give another proof of the fact that the structure space of each LBI-subalgebra is homeomorphic with a quotient of βX, which is proved in [15] via the mapping Z A . In Section 3, we consider the class of complete rings of functions which is introduced in [5]. It simply follows that every complete ring of functions is an LBI-subalgebra and thus the compactification associated with each complete ring of functions, which is identified in [5] via the mapping Z A , is just the structure space of that subring.…”

“…In Section 3, we consider the class of complete rings of functions which is introduced in [5]. It simply follows that every complete ring of functions is an LBI-subalgebra and thus the compactification associated with each complete ring of functions, which is identified in [5] via the mapping Z A , is just the structure space of that subring. Thus, some results of [5] could be achieved in a different way.…”