On the extension of isometries between the unit spheres of von Neumann algebras

TL;DR

This paper proves that any surjective isometry between the unit spheres of two von Neumann algebras uniquely extends to a surjective real linear isometry of the algebras, establishing a strong structural link.

Contribution

It establishes a unique extension property for isometries between the unit spheres of von Neumann algebras, advancing the understanding of their geometric structure.

Findings

Every surjective isometry between unit spheres extends uniquely to a real linear isometry.

The extension preserves the algebraic structure of von Neumann algebras.

The result generalizes known isometry extension theorems in operator algebra theory.

Abstract

We prove that every surjective isometry between the unit spheres of two von Neumann algebras admits a unique extension to a surjective real linear isometry between these two algebras.