On m-Closure of Ideals in LBI-Subalgebras

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发表日期：

2024年7月11日

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Masoumeh Etebar,
Mehdi Parsinia,
Alireza Salehi

摘要

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 LBI-subalgebra, if for every function f in A(X) that is bounded away from zero on a cozero-set E of X, there exists $g\in A(X)$ $g \in A (X)$ such that $fg|_E=1$ ${f g |}_{E} = 1$ . In this paper, for an LBI-subalgebra A(X) the compactification $\beta _AX$ $β_{A} X$ of X which is homeomorphic with the structure space of A(X) is investigated. Some properties of $\beta _AX$ $β_{A} X$ similar to the counterparts in $\beta X$ $β X$ and some main differences between these compactifications are given. Using the compactification $\beta _AX$ $β_{A} X$ , we establish an m-closure formula for ideals in a class of LBI-subalgebras which provides a generalization of m-closure of ideals in intermediate algebras of C(X) and $C_c(X)$ $C_{c} (X)$ . We also investigate a characterization of $\beta $ $β$ -ideals for LBI-subalgebras from which it turns out that m-closed ideals coincide with $\beta $ $β$ -ideals in that class of LBI-subalgebras.