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

Abstract

We show that every ring isomorphism between the algebras of locally measurable operators for type II 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.