Published July 11, 2024 | Version v1
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On m-Closure of Ideals in LBI-Subalgebras

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Let C(X) be the ring of all continuous real-valued functions on a completely regular Hausdorff space X. A subalgebra A(X) of C(X) is said to be closed under local bounded inversion, briefly an L BI-subalgebra, if for every function f in A(X) that is bounded away from zero on a cozero-set E of X, there exists g ∈ A(X) such that f g|E = 1. In this paper, for an L BI-subalgebra A(X) the compactification βA X of X which is homeomorphic with the structure space of A(X) is investigated. Some properties of βA X similar to the counterparts in β X and some main differences between these compactifications are given. Using the compactification βA X, we establish an m-closure formula for ideals in a class of L BI-subalgebras which provides a generalization of m-closure of ideals in intermediate algebras of C(X) and Cc(X). We also investigate a characterization of β-ideals for L BI-subalgebras from which it turns out that m-closed ideals coincide with β-ideals in that class of L BI-subalgebras.

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