A bilinear version of Holsztynski's theorem on isometries of C(X)-spaces

We prove that, for a compact metric space X not reduced to a point, the existence of a bilinear mapping circle: C(X) x C(X) -> C(X) satisfying //f circle g// = //f// //g// for all f, g is an element of C (X) is equivalent to the uncountability of X. This is derived from a bilinear version of Holsztynski's theorem [3] on isometries of C(X)-spaces, which is also proved in the paper.