Poly-adic filtrations, standardness, complementability and maximality

Abstract

Given some essentially separable filtration $(\mathcal{Z}_{n})_{n\le0}$ indexed by the nonpositive integers, we define the notion of complementability for the filtrations contained in $(\mathcal{Z}_{n})_{n\le0}$. We also define and characterize the notion of maximality for the poly-adic sub-filtrations of $(\mathcal{Z}_{n})_{n\le0}$. We show that any poly-adic sub-filtration of $(\mathcal{Z}_{n})_{n\le0}$ which can be complemented by a Kolmogorovian filtration is maximal in $(\mathcal{Z}_{n})_{n\le0}$. We show that the converse is false, and we prove a partial converse, which generalizes Vershik’s lacunary isomorphism theorem for poly-adic filtrations.