Vol. 68, No. 1, 1977

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Tensor products of function rings under composition

Nathan Jacob Fine

Vol. 68 (1977), No. 1, 63–72

Let C(X), C(Y ) be the rings of real-valued continuous functions on the completely regular Hausdorff spaces X, Y and let T = C(X) C(Y ) be the subring of C(X ×Y ) generated by functions of the form fg, where f C(X) and g C(Y ). If P is a real polynomial, then P t T for every t T. If G t T for all t T and if G is analytic, then G is a polynomial, provided that X and Y are both infinite (A. W. Hager, Math. Zeitschr. 92, (1966), 210–224, Prop. 3.). In this note I remove the condition of analyticity. Clearly the cardinality condition is necessary, for if either X or Y is finite, then T = C(X × Y ) and G t T for every continuous G and for every t T.

Mathematical Subject Classification 2000
Primary: 54C40
Secondary: 46M99
Received: 18 May 1976
Published: 1 January 1977
Nathan Jacob Fine