A bilinear version of Holsztyński's theorem on isometries of $C(X)$-spaces
Tom 166 / 2005
Studia Mathematica 166 (2005), 83-91
MSC: Primary 46E15, 46B04; Secondary 46H70.
DOI: 10.4064/sm166-1-6
Streszczenie
We prove that, for a compact metric space $X$ not reduced to a point, the existence of a bilinear mapping $\diamond :C(X)\times C(X)\to C(X)$ satisfying $\Vert f\diamond g\Vert =\Vert f\Vert\, \Vert g\Vert $ for all $f,g\in C(X)$ is equivalent to the uncountability of $X$. This is derived from a bilinear version of Holsztyński's theorem [3] on isometries of $C(X)$-spaces, which is also proved in the paper.
