Automorphisms of central extensions of type I von Neumann algebras

TL;DR

This paper studies automorphisms of the central extension of type I von Neumann algebras, showing they decompose into inner automorphisms and automorphisms from the center, with special results for type I_infinity.

Contribution

It provides a decomposition of automorphisms of the central extension of type I von Neumann algebras into inner and center-generated automorphisms, highlighting the structure of these automorphisms.

Findings

Automorphisms decompose into inner and center automorphisms.

Every band preserving automorphism of E(M) for type I_infinity is inner.

The structure of automorphisms is explicitly characterized for type I von Neumann algebras.

Abstract

Given a von Neumann algebra $M$ we consider the central extension $E(M)$ of $M.$ For type I von Neumann algebras $E(M)$ coincides with the algebra $LS(M)$ of all locally measurable operators affiliated with $M.$ In this case we show that an arbitrary automorphism $T$ of $E(M)$ can be decomposed as $T=T_a\circ T_\phi,$ where $T_a(x)=axa^{-1}$ is an inner automorphism implemented by an element $a\in E(M),$ and $T_\phi$ is a special automorphism generated by an automorphism $\phi$ of the center of $E(M).$ In particular if $M$ is of type I$_\infty$ then every band preserving automorphism of $E(M)$ is inner.