Valuations on Banach lattices

TL;DR

This paper develops a comprehensive framework for understanding valuations on Banach lattices, extending existing theories to include decomposition, boundedness, and integral representations of continuous valuations.

Contribution

It introduces a general approach to valuations on Banach lattices, expanding prior work on function spaces like $L_p(mu)$, Orlicz, and $C(K)$ spaces.

Findings

Characterization of decomposition properties of valuations

Results on boundedness of continuous valuations

Integral representation formulas for valuations

Abstract

We provide a general framework for the study of valuations on Banach lattices. This complements and expands several recent works about valuations on function spaces, including $L_p(\mu)$, Orlicz spaces and spaces $C(K)$ of continuous functions on a compact Hausdorff space. In particular, we study decomposition properties, boundedness and integral representation of continuous valuations.