ar X iv : m at h / 05 09 00 6 v 1 [ m at h . FA ] 1 S ep 2 00 5 ROOT CLOSED FUNCTION ALGEBRAS ON COMPACTA OF LARGE DIMENSION
نویسنده
چکیده
Let X be a Hausdorff compact space and C(X) be the algebra of all continuous complex-valued functions on X, endowed with the supremum norm. We say that C(X) is (approximately) n-th root closed if any function from 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 ndivisibility of first Čech cohomology groups of closed subsets of X. Next, for each positive integer m we construct m-dimensional metrizable compactum X such that C(X) is approximately n-th root closed for any n. Also, for each positive integer m we construct m-dimensional compact Hausdorff space X such that C(X) is n-th root closed for any n.
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