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

Abstract:

Given two Banach spaces $F||$ and $X|| ||$, 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 $||x| - ||Vx||| \leq \varepsilon$ 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.