On m-Closure of Ideals in LBI-Subalgebras

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 \(g\in A(X)\) such that \(fg|_E=1\). In this paper, for an LBI-subalgebra A(X) the compactification \(\beta _AX\) of X which is homeomorphic with the structure space of A(X) is investigated. Some properties of \(\beta _AX\) similar to the counterparts in \(\beta X\) and some main differences between these compactifications are given. Using the compactification \(\beta _AX\), 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 \(C_c(X)\). We also investigate a characterization of \(\beta \)-ideals for LBI-subalgebras from which it turns out that m-closed ideals coincide with \(\beta \)-ideals in that class of LBI-subalgebras.