## Banach Journal of Mathematical Analysis

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

#### Abstract

We prove that, if Banach spaces $X$ and $Y$ are $\delta$-average rough, then their direct sum with respect to an absolute norm $N$ is $\delta/N(1,1)$-average rough. In particular, for octahedral $X$ and $Y$ and for $p$ in $(1,\infty)$, the space $X\oplus_{p}Y$ is $2^{1-1/p}$-average rough, which is in general optimal. Another consequence is that for any $\delta$ in $(1,2]$ there is a Banach space which is exactly $\delta$-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 $\infty$-sums.

#### Article information

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

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

https://projecteuclid.org/euclid.bjma/1512464420

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

Mathematical Reviews number (MathSciNet)
MR3745582

Zentralblatt MATH identifier
06841273

#### 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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