Non-ergodic Banach spaces are near Hilbert

Cuellar Carrera, W.

doi:10.1090/tran/7319

Non-ergodic Banach spaces are near Hilbert
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by W. Cuellar Carrera PDF

Trans. Amer. Math. Soc. 370 (2018), 8691-8707 Request permission

Abstract:

We prove that a non-ergodic Banach space must be near Hilbert. In particular, $\ell _p$ ($2<p<\infty$) is ergodic. This reinforces the conjecture that $\ell _2$ is the only non-ergodic Banach space. As an application of our criterion for ergodicity, we prove that there is no separable Banach space which is complementably universal for the class of all subspaces of $\ell _p$, for $1\leq p <2$. This solves a question left open by W. B. Johnson and A. Szankowski in 1976.

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