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A class of subalgebras of $$C(X)$$ and the associated compactness. (English) Zbl 1012.54024
Let $$X$$ be a Tikhonov space, $$C(X)$$ the ring of continuous real valued functions on $$X$$, and $$C^*(X)$$ the bounded members of $$C(X)$$. A result of D. Plank [Fundam. Math. 64, 41-54 (1969; Zbl 0182.56302)] says that if $$A\supset C^*(X)$$ is a subalgebra of $$C(X)$$, then the Stone-Čech compactification $$\beta X$$ can be obtained as the set of maximal ideals of $$A$$ with the hull-kernel topology. The subspace of real maximal ideals in $$A$$, denoted by $$\upsilon_AX$$, is called the $$A$$-compactification of $$X$$; $$X$$ is called $$A$$-compact in case $$\upsilon_AX=X.$$ A result of I. Redlin and S. Watson [Proc. Am. Math. Soc. 100, 763-766 (1987; Zbl 0622.54011)] says that if $$B\supset C^*(Y)$$ is a subalgebra of $$C(Y)$$, $$X$$ is $$A$$-compact, $$Y$$ is $$B$$-compact, and $$A$$ is isomorphic to $$B$$, then $$X$$ is homeomorphic to $$Y$$. The converse, that (i) $$X$$ homeomorphic to $$Y$$ implies $$A$$ isomorphic to $$B$$, is not valid in general, as they show by example. In the present paper it is shown that (i) is valid, provided $$A$$ satisfies the condition that (ii) if $$f\in C(X)$$ has an extension over $$\upsilon_AX$$, then $$f\in A$$; and provided $$B$$ satisfies its version of (ii). They give an example of $$A\supset C^*(X)$$ that does not satisfy (ii). In case $$Y = X$$, they derive conditions which are equivalent to $$\upsilon_AX = \upsilon_BX$$. Several open questions are also posed.

##### MSC:
 54C40 Algebraic properties of function spaces in general topology 46E25 Rings and algebras of continuous, differentiable or analytic functions 46H10 Ideals and subalgebras 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54D60 Realcompactness and realcompactification