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Weakly stable Banach spaces. (English) Zbl 0639.46018
A separable Banach space is called stable if for every pair of sequences $$\{x_ m\}$$ and $$\{y_ n\}$$ and any two ultrafilters $$U$$ and $$V$$ on $$\mathbb{N}$$ the equality $\lim_{m,U}\lim_{n,V}\| x_ m+y_ n\| = \lim_{n,V}\lim_{m,U}\| x_ m+y_ n\|$ holds. [Equivalently, for such sequences $$\sup_{m>n}\| x_ m-y_ n\| >\inf_{m<n}\| x_ m-y_ n\|$$ see D. J. H. Garling, Lect. Notes Math. 928, 121-175 (1982; Zbl 0485.46012)].
In this paper the authors generalize this notion by defining a weakly stable space to be as above, but only requiring the sequences $$\{x_ m\}$$ and $$\{y_ m\}$$ to be in an arbitrary weakly compact set. They obtain a generalization of a theorem of J. L. Krivine and B. Maurey [Isr. J. Math. 39, 273-295 (1981; Zbl 0504.46013)] concerning stable spaces by showing that if $$X$$ is weakly stable then it contains a subspace $$(1+\epsilon)$$-isomorphic to either an $$\ell_p$$-space or to $$c_ 0$$ (for any $$\epsilon >0)$$. In this theorem the possibility of the subspace being isomorphic to $$c_ 0$$ is what generalizes the Krivine-Maurey result; however the authors show that the concept of weak stability properly generalizes that of stability even in the class of spaces not contaning $$c_ 0$$ by constructing a weakly stable space which is not stable under any equivalent norm, but with the property that any infinite dimensional subspace contains $$\ell_2$$. It is also shown that the $$\ell_p$$-sum of weakly stable spaces is weakly stable, but that this need not be true for the $$c_ 0$$-sum.
Reviewer: J.R.Holub

##### MSC:
 46B20 Geometry and structure of normed linear spaces 46B25 Classical Banach spaces in the general theory
##### Keywords:
ultrafilters; $$\ell_p$$-sum; weakly stable spaces
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##### References:
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