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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Root closed function algebras on compacta of large dimension


Authors: N. Brodskiy, J. Dydak, A. Karasev and K. Kawamura
Journal: Proc. Amer. Math. Soc. 135 (2007), 587-596
MSC (2000): Primary 54F45; Secondary 46J10
Published electronically: August 8, 2006
MathSciNet review: 2255306
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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 Cech 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$.


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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
Email: kawamura@math.tsukuba.as.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08490-5
PII: S 0002-9939(06)08490-5
Keywords: Algebraically closed algebras, approximately root closed algebras, commutative Banach algebras, dimension
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
Article copyright: © Copyright 2006 American Mathematical Society