Root closed function algebras on compacta of large dimension

Abstract

Let X X be a Hausdorff compact space and let C ( X ) C(X) be the algebra of all continuous complex-valued functions on X X , endowed with the supremum norm. We say that C ( X ) C(X) is (approximately) n n -th root closed if any function from C ( X ) C(X) is (approximately) equal to the n n -th power of another function. We characterize the approximate n n -th root closedness of C ( X ) C(X) in terms of n n -divisibility of the first Čech cohomology groups of closed subsets of X X . Next, for each positive integer m m we construct an m m -dimensional metrizable compactum X X such that C ( X ) C(X) is approximately n n -th root closed for any n n . Also, for each positive integer m m we construct an m m -dimensional compact Hausdorff space X X such that C ( X ) C(X) is n n -th root closed for any n n .