Renorming AM-spaces

TL;DR

This paper demonstrates that separable AM-spaces can be renormed to eliminate non-trivial surjective lattice isometries, introducing Benyamini spaces as a key tool for approximation.

Contribution

It introduces a method to renorm separable AM-spaces to remove non-trivial lattice isometries and develops the concept of Benyamini spaces for approximation.

Findings

Existence of equivalent lattice norms with trivial isometry groups

Construction of AM-norms for spaces with at most one atom

Introduction and analysis of Benyamini spaces

Abstract

We prove that any separable AM-space $X$ has an equivalent lattice norm for which no non-trivial surjective lattice isometries exist. Moreover, if $X$ has no more than one atom, then this new norm may be an AM-norm. As our main tool, we introduce and investigate the class of so called Benyamini spaces, which ``approximate'' general AM-spaces.