A note on inverse-closed subalgebras of C(X)
The word space will refer to Tychonoff spaces. We denote C(X) for the ring of all continuous real-valued functions on a space X. By an algebra on X is meant a subring of C(X) which contains the constants, separate points and closed sets, and is closed under uniform convergence and inversion in C(X). A. W. Hager and D. G. Johnson have proved that if the Hewitt realcompactification \(\upsilon\) X is Lindelöf, then each algebra on X is isomorphic to C(Y) for some space Y and they ask whether the converse of this assertion holds. In an earlier paper [Stud. Sci. Math. Hung. (to appear); Zbl 0526.54014] the author has proved that the answer is yes if X is paracompact or X is the product \(Y\times K\), where K is a compact space of non-measurable cardinal having at least two points. In this paper we provide another partial solution to the above question by showing the equivalence when X is a Hausdorff topological group.
scientific article; zbMATH DE number 3899724