Isometries between projection lattices of von Neumann algebras

TL;DR

This paper characterizes isometries between projection lattices of von Neumann algebras, showing they correspond to Jordan *-isomorphisms, and extends previous results to a broader class of algebras.

Contribution

It establishes a new characterization of isometries between projection lattices via Jordan *-isomorphisms for a wide class of von Neumann algebras.

Findings

Isometries correspond to Jordan *-isomorphisms.

Two von Neumann algebras are Jordan *-isomorphic iff their projection lattices are isometric.

Extends previous results from type I factors to more general algebras.

Abstract

We investigate surjective isometries between projection lattices of two von Neumann algebras. We show that such a mapping is characterized by means of Jordan $^*$-isomorphisms. In particular, we prove that two von Neumann algebras without type I$_1$ direct summands are Jordan $^*$-isomorphic if and only if their projection lattices are isometric. Our theorem extends the recent result for type I factors by G.P. Geh\'er and P. \v{S}emrl, which is a generalization of Wigner's theorem.