Extreme point methods in the study of isometries on certain non-commutative spaces

TL;DR

This paper characterizes surjective isometries on specific non-commutative spaces linked to semi-finite von Neumann algebras, using extreme point techniques to obtain global representations.

Contribution

It provides the first comprehensive characterization of isometries on these non-commutative Lorentz and related spaces, extending previous results.

Findings

Characterization of surjective isometries on Lorentz spaces $L^{w,1}$

Representation of isometries as global transformations

Use of extreme points of unit balls for characterization

Abstract

In this paper we characterize surjective isometries on certain classes of non-commutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces $L^{w,1}$, as well as the spaces $L^1+L^\infty$ and $L^1\cap L^\infty$. The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces. Of particular interest is that the representations of isometries obtained in this paper are global representations.