The ergodicity of weak Hilbert spaces.

This paper complements a recent result of Dilworth, Ferenczi, Kutzarova and Odell regarding the ergodicity of strongly asymptotic $ell _p$ 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 $ell _2$ is ergodic. In particular, every weak Hilbert space which is not isomorphic to $ell _2$ must be ergodic. Throughout the paper we construct explicitly the maps which establish the fact that the relation $E_0$ is Borel reducible to isomorphism between subspaces of the Banach spaces involved. [ABSTRACT FROM AUTHOR]