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Zbl 1150.46003
Becerra Guerrero, Julio; Rodriguez Palacios, Angel
Characterizations of almost transitive superreflexive Banach spaces. (English)
[J] Commentat. Math. Univ. Carol. 42, No. 4, 629-636 (2001). ISSN 0010-2628

A Banach space $X$ is almost transitive if for some $x\in X$ of norm 1 the set $G$ of images of $x$ under all surjective linear isometries of $X$ is dense in the unit sphere of $X$. It is convex transitive if the convex hull of $G$ is dense in the unit ball of $X$. According to {\it C. Finet} [Isr. J. Math. 53, 81--92 (1986; Zbl 0603.46016)], almost transitive spaces are uniformly convex and uniformly smooth. The authors characterize them in terms of ``uniform Fréchet non-differentiability'' of the norm. A Banach space $X$ is almost transitive if and only if $X$ is convex transitive and either $X$ or $X^*$ is not extremely rough.
[Eva Matoušková (Praha)]

MSC 2000:: *46B04 Isometric theory of Banach spaces
46B10 Duality and reflexivity in normed spaces
46B22 Spaces with Radon-Nikodym property

Keywords: uniformly convex space; uniformly smooth space; rough space; surjective isometry; almost transitive Banach space

Citations: Zbl 0603.46016

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