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Convexity and smoothness of Banach spaces with numerical index one. (English) Zbl 1197.46008

The authors study whether smoothness or convexity conditions are compatible with isometric Banach space notions such as the (alternative) Daugavet property, having numerical index \(1\) or lushness, which was introduced in [K. Boyko, V. Kadets, M. Martín and D. Werner, Math. Proc. Camb. Philos. Soc. 142, No. 1, 93–102 (2007; Zbl 1121.47001)]. A sample of their results is as follows. In the following, we tacitly assume that all Banach spaces are of dimension \(>1\). 7mm
(1)
If \(X\) has the alternative Daugavet property, then \(X^*\) is neither smooth nor strictly concex, and \(X\) is not Fréchet smooth.
(2)
There is a strictly convex non-complete normed space that is lush; in particular, it has numerical index \(1\).
(3)
However, a real lush Banach space is neither smooth nor strictly convex; the complex case remains open.
(4)
Let \(X\) be a closed smooth or strictly convex subspace of the real space \(C[0,1]\), then \(C[0,1]/X\) contains a copy of \(C[0,1]\). This follows from (3) and the general theory of rich subspaces.
(5)
If \(X\) is an infinite-dimensional real lush space, then \(\ell_1\) embeds into \(X^*\).
There are many more results, for example on extreme points of lush spaces, in this interesting and very well written paper.

MSC:

46B04 Isometric theory of Banach spaces
46B20 Geometry and structure of normed linear spaces
47A12 Numerical range, numerical radius

Citations:

Zbl 1121.47001
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Full Text: arXiv Euclid

References:

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