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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
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