THE ROOT PROBLEM IN COMPLEX F-ALGEBRAS

Abstract

It is well known that the square root of a product of two positive elements exists in an Archimedean, uniformly complete semiprime f-algebra. In particular, if the f-algebra is unital, then the square root of any positive element exists. It is natural to ask whether the square root of every element exits in the complexification of such an f-algebra. In this note we show that the root does not exist in general. Some conditions for the existence of the root are given.