Root closed function algebras on compacta of large dimension

N. BRODSKIY, J. DYDAK, A. KARASEV, AND K. KAWAMURAAbstract. Let X be a Hausdorff compact space and C(X) be the algebra ofall continuous complex-valued functions on X, endowed with the supremumnorm. We say that C(X) is (approximately) n-th root closed if any functionfrom C(X) is (approximately) equal to the n-th power of another function.We characterize the approximate n-th root closedness of C(X) in terms of n-divisibility of first Cech cohomology groups of closed subsets ofˇ X. Next, foreach positive integer m we construct m-dimensional metrizable compactum Xsuch that C(X) is approximately n-th root closed for any n. Also, for eachpositive integer m we construct m-dimensional compact Hausdorff space Xsuch that C(X) is n-th root closed for any n.