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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On $ J$-convexity and some ergodic super-properties of Banach spaces

Authors: Antoine Brunel and Louis Sucheston
Journal: Trans. Amer. Math. Soc. 204 (1975), 79-90
MSC: Primary 46B05
MathSciNet review: 0380361
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Abstract: Given two Banach spaces $ F\vert\vert$ and $ X\vert\vert\,\vert\vert$, write $ F{\text{ fr }}X{\text{ iff}}$ for each finite-dimensional subspace $ F'$ of $ F$ and each number $ \varepsilon > 0$, there is an isomorphism $ V$ of $ F'$ into $ X$ such that $ \vert\vert x\vert - \vert\vert Vx\vert\vert\vert \leq \varepsilon $ for each $ x$ in the unit ball of $ F'$. Given a property $ {\mathbf{P}}$ of Banach spaces, $ X$ is called super- $ {\mathbf{P}}{\text{ iff }}F{\text{ fr }}X$ implies $ F$ is $ {\mathbf{P}}$. Ergodicity and stability were defined in our articles On $ B$-convex Banach spaces, Math. Systems Theory 7 (1974), 294-299, and C. R. Acad. Sci. Paris Ser. A 275 (1972), 993, where it is shown that super-ergodicity and super-stability are equivalent to super-reflexivity introduced by R. C. James [Canad. J. Math. 24 (1972), 896-904]. $ Q$-ergodicity is defined, and it is proved that super-$ Q$-ergodicity is another property equivalent with super-reflexivity. A new proof is given of the theorem that $ J$-spaces are reflexive [Schaffer-Sundaresan, Math. Ann. 184 (1970), 163-168]. It is shown that if a Banach space $ X$ is $ B$-convex, then each bounded sequence in $ X$ contains a subsequence $ ({y_n})$ such that the Cesàro averages of $ {( - 1)^i}{y_i}$ converge to zero.

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Keywords: Super-properties, $ J$-convexity, $ B$-convexity, ergodic, reflexive, Banach-Saks
Article copyright: © Copyright 1975 American Mathematical Society