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Hardtke, Jan-David

Absolute sums of Banach spaces and some geometric properties related to rotundity and smoothness. (English) Zbl 1284.46012

Banach J. Math. Anal. 8, No. 1, 295-334 (2014).
The author studies in depth the notions of acs, luacs and uacs Banach spaces that form common generalizations of the usual rotundity and smoothness properties of Banach spaces. A Banach space \(X\) is called acs (alternatively convex or smooth) if, for every \(x, y \in S_X\) with \(\|x + y\| = 2\) and every \(x^* \in S_{X^*}\) with \(x^*(x) = 1\), one has \(x^*(y) = 1\) as well. \(X\) is called luacs (locally uniformly alternatively convex or smooth) if, for every \(x \in S_X\), every sequence \((x_n) \subset S_X\) and every functional \(x^* \in S_{X^*}\), the conditions \(\|x_n + y\| \to 2\) and \(x^*(x_n) \to 1\) imply that \(x^*(y) = 1\). Finally, \(X\) is called uacs (uniformly alternatively convex or smooth) if, for every triple of sequences \((x_n), (y_n)\subset S_X\), \((x_n^*) \subset S_{X^*}\), the conditions \(\|x_n + y_n\| \to 2\) and \(x_n^*(x_n) \to 1\) imply that \(x_n^*(y_n) \to 1\).
The asc spaces were introduced by V. M. Kadets [Quaest. Math. 19, No. 1–2, 225–235 (1996; Zbl 0855.47021)], luacs and uacs spaces by V. M. Kadets et al. [Trans. Am. Math. Soc. 352, No. 2, 855–873 (2000; Zbl 0938.46016)] as geometric tools in the study of the anti-Daugavet property.
The author gives a number of useful reformulations and collects relations with other geometric properties, in particular, with the properties sluacs and wuacs introduced in the paper under review. A large part of the paper is devoted to absolute sums of acs, luacs, uacs, sluacs and wuacs spaces.
Reviewer: Vladimir Kadets (Kharkov)

Cited in 1 Review
Cited in 6 Documents

MSC:

46B04 Isometric theory of Banach spaces
46B20 Geometry and structure of normed linear spaces

Keywords:

rotundity; acs spaces; uacs spaces; absolute sums; smoothness

Citations:

Zbl 0855.47021; Zbl 0938.46016
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\textit{J.-D. Hardtke}, Banach J. Math. Anal. 8, No. 1, 295--334 (2014; Zbl 1284.46012)
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