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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.

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