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

Abstract

Let $A(X)$ denote a subalgebra of $C(X)$ which is closed under local bounded inversion, briefly, an $LBI$-subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151--163. By characterizing maximal ideals of $A(X)$, we generalize the notion of $z_A^\beta$-ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of $C(X)$ containing $C^*(X)$, Comment. Math. Univ. Carolin. 48 (2007), 273--280 for intermediate subalgebras, to the $LBI$-subalgebras. Using these, it is simply shown that the structure space of every $LBI$-subalgebra is homeomorphic with a quotient of $\beta X$. This gives a different approach to the results of Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151--163 and also shows that the Banaschewski-compactification of a zero-dimensional space $X$ is a quotient of $\beta X$. Finally, we consider the class of complete rings of functions which was first defined in Byun H.L., Redlin L., Watson S., Local invertibility in subrings of $C^*(X)$, Bull. Austral. Math. Soc. 46 (1992), 449--458. Showing that every such subring is an $LBI$-subalgebra, we prove that the compactification of $X$ associated to each complete ring of functions, which is identified in Byun H.L., Redlin L., Watson S., Local invertibility in subrings of $C^*(X)$, Bull. Austral. Math. Soc. 46 (1992), 449--458 via the mapping ${\mathcal Z}_A$, is in fact, the structure space of that subring. Henceforth, some statements in Byun H.L., Redlin L., Watson S., Local invertibility in subrings of $C^*(X)$, Bull. Austral. Math. Soc. 46 (1992), 449--458 could be proved in a different way.