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The transfer of property $ (\beta)$ of Rolewicz by a uniform quotient map


Authors: S. J. Dilworth, Denka Kutzarova and N. Lovasoa Randrianarivony
Journal: Trans. Amer. Math. Soc. 368 (2016), 6253-6270
MSC (2010): Primary 46B80, 46B20, 46T99, 51F99
DOI: https://doi.org/10.1090/tran/6553
Published electronically: December 9, 2015
MathSciNet review: 3461033
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Abstract: We provide a Laakso construction to prove that the property of having an equivalent norm with the property $ (\beta )$ of Rolewicz is qualitatively preserved via surjective uniform quotient mappings between separable Banach spaces. On the other hand, we show that the $ (\beta )$-modulus is not quantitatively preserved via such a map by exhibiting two uniformly homeomorphic Banach spaces that do not have $ (\beta )$-moduli of the same power type even under renorming.


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Additional Information

S. J. Dilworth
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: dilworth@math.sc.edu

Denka Kutzarova
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
Email: denka@math.uiuc.edu

N. Lovasoa Randrianarivony
Affiliation: Department of Mathematics and Computer Science, Saint Louis University, St. Louis, Missouri 63103
Email: nrandria@slu.edu

DOI: https://doi.org/10.1090/tran/6553
Keywords: Property $(\beta)$ of Rolewicz, uniform quotient, Lipschtiz quotient, Laakso construction
Received by editor(s): August 23, 2013
Received by editor(s) in revised form: June 1, 2014, and August 11, 2014
Published electronically: December 9, 2015
Additional Notes: The first author was partially supported by NSF grant DMS–1101490. The third author was partially supported by NSF grant DMS–1301591.
Article copyright: © Copyright 2015 by the authors

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