Lattice isomorphisms between projection lattices of von Neumann algebras

TL;DR

This paper characterizes lattice isomorphisms between projection lattices of von Neumann algebras using ring isomorphisms of locally measurable operators, extending previous results and providing a full description in certain cases.

Contribution

It generalizes von Neumann's classical result to arbitrary von Neumann algebras and describes ring isomorphisms of locally measurable operators without type II summands.

Findings

Lattice isomorphisms correspond to ring isomorphisms of locally measurable operators.

Complete description of ring isomorphisms for algebras without type II summands.

Extension of von Neumann's result to broader classes of algebras.

Abstract

Generalizing von Neumann's result on type II$_1$ von Neumann algebras, we characterize lattice isomorphisms between projection lattices of arbitrary von Neumann algebras by means of ring isomorphisms between the algebras of locally measurable operators. Moreover, we give a complete description of ring isomorphisms of locally measurable operator algebras when the von Neumann algebras are without type II direct summands.