Ergodic Banach Spaces

Abstract

We show that any Banach space contains a continuum of non isomorphic subspaces or a minimal subspace. We define an ergodic Banach space $X$ as a space such that $E_0$ Borel reduces to isomorphism on the set of subspaces of $X$, and show that every Banach space is either ergodic or contains a subspace with an unconditional basis $whichiscomplementablyuniversalforthefamilyofitsblock-subspaces.Wealsouseourmethodstogetuniformityresults;forexample,incombinationwitharesultofB.Maurey,V.MilmanandN.Tomczak-Jaegermann,weshowthatanunconditionalbasisofaBanachspace,ofwhicheveryblock-subspaceiscomplemented,mustbeasymptotically$c_0$or$l_p$.