The transfer of property (𝛽) of Rolewicz by a uniform quotient map

Dilworth, S.; Kutzarova, Denka; Randrianarivony, N.

doi:10.1090/tran/6553

The transfer of property $(\beta )$ of Rolewicz by a uniform quotient map
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by S. J. Dilworth, Denka Kutzarova and N. Lovasoa Randrianarivony PDF

Trans. Amer. Math. Soc. 368 (2016), 6253-6270

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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