Fibers of $L^{\infty }$ algebra

Abstract

It is shown that Gelfand transforms of elements $f\in\lmu$ are constant at almost every fiber ${\Pi }^{-1}(\{x\})$ of the spectrum of $\lmu$ in the following sense: for each $f\in\lmu$ there is an open dense subset $U=U(f)$ of this spectrum having full measure and such that the Gelfand transform of $f$ is constant on the intersection ${\Pi }^{-1}(\{x\})\cap U$.

4 pages