Banach Journal of Mathematical Analysis

Stability of average roughness, octahedrality, and strong diameter 2 properties of Banach spaces with respect to absolute sums

Rainis Haller, Johann Langemets, and Rihhard Nadel

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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.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 1 (2018), 222-239.

Dates
Received: 10 February 2017
Accepted: 15 May 2017
First available in Project Euclid: 5 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1512464420

Digital Object Identifier
doi:10.1215/17358787-2017-0040

Mathematical Reviews number (MathSciNet)
MR3745582

Zentralblatt MATH identifier
06841273

Subjects
Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 46B22: Radon-Nikodým, Kreĭn-Milman and related properties [See also 46G10]

Keywords
average rough norm octahedral norm diameter 2 property Daugavet property

Citation

Haller, Rainis; Langemets, Johann; Nadel, Rihhard. Stability of average roughness, octahedrality, and strong diameter $2$ properties of Banach spaces with respect to absolute sums. Banach J. Math. Anal. 12 (2018), no. 1, 222--239. doi:10.1215/17358787-2017-0040. https://projecteuclid.org/euclid.bjma/1512464420


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