Compact groups of positive operators on Banach lattices

TL;DR

This paper investigates the structure of groups of positive operators on Banach lattices, providing characterizations and decompositions under compactness and factorization conditions, with applications to function spaces.

Contribution

It introduces a new factorization property leading to isometric representations and characterizes positive group representations with compact images.

Findings

Groups with the factorization property have isomorphic images in isometric positive operators.

Compact groups of positive operators are conjugate to isometric groups via a central automorphism.

An ordered decomposition theorem for unitary representations is established for symmetric Banach sequence spaces.

Abstract

In this paper we study groups of positive operators on Banach lattices. If a certain factorization property holds for the elements of such a group, the group has a homomorphic image in the isometric positive operators which has the same invariant ideals as the original group. If the group is compact in the strong operator topology, it equals a group of isometric positive operators conjugated by a single central lattice automorphism, provided an additional technical assumption is satisfied, for which we again have only examples. We obtain a characterization of positive representations of a group with compact image in the strong operator topology, and use this for normalized symmetric Banach sequence spaces to prove an ordered version of the decomposition theorem for unitary representations of compact groups. Applications concerning spaces of continuous functions are also considered.