The ergodicity of weak Hilbert spaces

Abstract

This paper complements a recent result of Dilworth, Ferenczi, Kutzarova and Odell regarding the ergodicity of strongly asymptotic ℓ p \ell _p spaces. We state this result in a more general form, involving domination relations, and we show that every asymptotically Hilbertian space which is not isomorphic to ℓ 2 \ell _2 is ergodic. In particular, every weak Hilbert space which is not isomorphic to ℓ 2 \ell _2 must be ergodic. Throughout the paper we construct explicitly the maps which establish the fact that the relation E 0 E_0 is Borel reducible to isomorphism between subspaces of the Banach spaces involved.