A characterization of orthogonal vector fields over ${\bf W^*}$-algebras of type ${\bf I_2}$

TL;DR

This paper characterizes $w^*$-continuous orthogonal vector fields over type $I_2$ $W^*$-algebras, showing they are determined by their reductions on the center and are stationary.

Contribution

It provides a new characterization of orthogonal vector fields over type $I_2$ $W^*$-algebras using reductions on the center, and proves their stationarity.

Findings

Characterization of orthogonal vector fields via reductions on the center

Proof that such fields are stationary over type $I_2$ $W^*$-algebras

Application of the characterization to establish stationarity

Abstract

In the paper we give a characterization of a $w^*$-continuous orthogonal vector field $F$ over an $W^*$-algebra $\mathcal{N}$ of type $I_2$ in terms of reductions $F$ on the center of $\mathcal{N}$. As an application it is obtained a proof of the assertion that an arbitrary $w^*$-continuous orthogonal vector field over a $W^*$-algebra of type $I_2$ is stationary.