On the existence of continuous (approximate) roots of algebraic equations

Root closed function algebras on compacta of large dimension

“…In this paper we characterize the approximate n-th root closedness of C(X) for any Hausdorff paracompact space X. Namely, C(X) is approximately n-th root closed if and only if the group H 1 (A; Z) is n-divisible for every closed subset A of X. If dimX ≤ 1, then the n-divisibility of H 1 (X; Z) implies the n-divisibility of H 1 (A; Z), so this generalizes Theorem 1.3 of [10]. Further, 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. Note that such examples were known in dimension 1 only.…”

“…Namely, in this case C(X) is algebraically closed if and only if X is a dendrite (i.e. a Peano continuum containing no simple closed curves) [10,13].…”

“…Further, 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. Note that such examples were known in dimension 1 only. 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. This example solves the problem posed in [10]: for a compact Hausdorff space X, does square root closedness of C(X) imply dimX ≤ 1?…”

OnC*-algebras with the approximaten-th root property

“…We say that the space C*(X) of all bounded complex-valued functions on X has the approximate n-th root property if X satisfies condition (*)". The results in this paper were inspired by the following theorem established by Kawamura and Miura [10]: THEOREM 1 . 1 .…”

“…Section 4 deals with square root closed compacta, compacta X such that, for every f ∈ C(X), there is g ∈ C(X) with f = g 2 . It is known that if X is a firstcountable connected compactum, then X is square-root closed if and only if X is locally connected, dim X ≤ 1 and Ȟ1 (X; Z) is trivial, see [6], [8], [10] and [12]. A topological characterization of general square root closed compacta has not been known.…”