Ring isomorphisms of type II$_\infty$ locally measurable operator algebras

TL;DR

This paper characterizes ring isomorphisms between algebras of locally measurable operators for type II$_ finite$ von Neumann algebras, showing they are similar to real *-isomorphisms, extending previous classifications.

Contribution

It provides a complete description of ring isomorphisms for these operator algebras, linking them to real *-isomorphisms and generalizing prior results.

Findings

Ring isomorphisms are similar to real *-isomorphisms.

Complete classification of isomorphisms for locally measurable operator algebras.

Extension of previous results to broader class of von Neumann algebras.

Abstract

We show that every ring isomorphism between the algebras of locally measurable operators for type II$_\infty$ von Neumann algebras is similar to a real $^*$-isomorphism. This together with previous results by the author and Ayupov--Kudaybergenov completely describes ring isomorphisms between the algebras of locally measurable operators as well as lattice isomorphisms between the projection lattices for a general pair of von Neumann algebras without finite type I direct summands.