Root closed function algebras on compacta of large dimension
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- by N. Brodskiy, J. Dydak, A. Karasev and K. Kawamura PDF
- Proc. Amer. Math. Soc. 135 (2007), 587-596 Request permission
Abstract:
Let $X$ be a Hausdorff compact space and let $C(X)$ 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 $n$-divisibility of the first Čech cohomology groups of closed subsets of $X$. Next, for each positive integer $m$ we construct an $m$-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 an $m$-dimensional compact Hausdorff space $X$ such that $C(X)$ is $n$-th root closed for any $n$.References
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Additional Information
- N. Brodskiy
- Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
- Email: brodskiy@math.utk.edu
- J. Dydak
- Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
- Email: dydak@math.utk.edu
- A. Karasev
- Affiliation: Department of Mathematics, Nipissing University, North Bay, Ontario, Canada P1B 8L7
- Email: alexandk@nipissingu.ca
- K. Kawamura
- Affiliation: Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8071, Japan
- MR Author ID: 255566
- Email: kawamura@math.tsukuba.as.jp
- Received by editor(s): June 5, 2005
- Received by editor(s) in revised form: August 31, 2005
- Published electronically: August 8, 2006
- Additional Notes: The third author was partially supported by an NSERC Grant.
- Communicated by: Alexander N. Dranishnikov
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 587-596
- MSC (2000): Primary 54F45; Secondary 46J10
- DOI: https://doi.org/10.1090/S0002-9939-06-08490-5
- MathSciNet review: 2255306