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

DOI:
https://doi.org/10.1090/S0002-9939-06-08490-5

Published electronically:
August 8, 2006

MathSciNet review:
2255306

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a Hausdorff compact space and let be the algebra of all continuous complex-valued functions on , endowed with the supremum norm. We say that is (approximately) -th root closed if any function from is (approximately) equal to the -th power of another function. We characterize the approximate -th root closedness of in terms of -divisibility of the first Cech cohomology groups of closed subsets of . Next, for each positive integer we construct an -dimensional metrizable compactum such that is approximately -th root closed for any . Also, for each positive integer we construct an -dimensional compact Hausdorff space such that is -th root closed for any .

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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:
https://doi.org/10.1090/S0002-9939-06-08490-5

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