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Characterization of function rings between \(C^*(X)\) and \(C(X)\). (English) Zbl 1120.54014
Summary: Let \(X\) be a Tychonoff space and \(\sum(X)\) the set of all the subrings of \(C(X)\) that contain \(C^*(X)\). For any \(A(X)\) in \(\sum(X)\) suppose \(v_AX\) is the largest sub-space of \(\beta X\) containing \(X\) to which each function in \(A(X)\) can be extended continuously. Let us write \(A(X)\sim B(X)\) if and only if \(v_AX=v_BX\), thereby defining an equivalence relation on \(\sum(X)\). We show that an \(A(X)\) in \(\sum(X)\) is isomorphic to \(C(Y)\) for some space \(Y\) if and only if \(A(X)\) is the largest member of its equivalence class if and only if there exists a subspace \(T\) of \(\beta X\) with the property that \(A(X)=\{f\in C(X):f^* (p)\) is real for each \(p\) in \(T\}\), \(f^*\) being the unique continuous extension of \(f\) in \(C(X)\) from \(\beta X\) to \(\mathbb{R}^*\), the one point compactification of \(\mathbb{R}\). As a consequence it follows that if \(X\) is a realcompact space in which every \(C^*\)-embedded subset is closed, then \(C(X)\) is never isomorphic to any \(A(X)\) in \(\sum(X)\) without being equal to it.

54C40 Algebraic properties of function spaces in general topology
54C35 Function spaces in general topology