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${}_{\infty }$ von Neumann algebras is similar to a real ${}^{\ast }$-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.

arXiv comment14 pages