# Isometries between non-commutative symmetric spaces associated with semi-finite von Neumann algebras

## TL;DR

This paper characterizes the structure of positive surjective isometries between symmetric spaces linked to semi-finite von Neumann algebras, revealing conditions under which they preserve disjointness and describing their form.

## Contribution

It provides a new structural description of isometries in non-commutative symmetric spaces, extending previous results by removing positivity assumptions in certain cases.

## Key findings

- Positive surjective isometries are projection disjointness preserving under finiteness conditions.
- Structural descriptions of these isometries are obtained.
- Results apply to strongly symmetric spaces with absolutely continuous norms.

## Abstract

In this article we show that positive surjective isometries between symmetric spaces associated with semi-finite von Neumann algebras are projection disjointness preserving if they are finiteness preserving. This is subsequently used to obtain a structural description of such isometries. Furthermore, it is shown that if the initial symmetric space is a strongly symmetric space with absolutely continuous norm, then a similar structural description can be obtained without requiring positivity of the isometry.