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January 2018 Stability of average roughness, octahedrality, and strong diameter 2 properties of Banach spaces with respect to absolute sums
Rainis Haller, Johann Langemets, Rihhard Nadel
Banach J. Math. Anal. 12(1): 222-239 (January 2018). DOI: 10.1215/17358787-2017-0040

Abstract

We prove that, if Banach spaces X and Y are δ-average rough, then their direct sum with respect to an absolute norm N is δ/N(1,1)-average rough. In particular, for octahedral X and Y and for p in (1,), the space XpY is 211/p-average rough, which is in general optimal. Another consequence is that for any δ in (1,2] there is a Banach space which is exactly δ-average rough. We give a complete characterization when an absolute sum of two Banach spaces is octahedral or has the strong diameter 2 property. However, among all of the absolute sums, the diametral strong diameter 2 property is stable only for 1- and -sums.

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Rainis Haller. Johann Langemets. Rihhard Nadel. "Stability of average roughness, octahedrality, and strong diameter 2 properties of Banach spaces with respect to absolute sums." Banach J. Math. Anal. 12 (1) 222 - 239, January 2018. https://doi.org/10.1215/17358787-2017-0040

Information

Received: 10 February 2017; Accepted: 15 May 2017; Published: January 2018
First available in Project Euclid: 5 December 2017

zbMATH: 06841273
MathSciNet: MR3745582
Digital Object Identifier: 10.1215/17358787-2017-0040

Subjects:
Primary: 46B20
Secondary: 46B22

Rights: Copyright © 2018 Tusi Mathematical Research Group

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Vol.12 • No. 1 • January 2018
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