A characterization of orthogonal vector fields over ${\mathbf{W}}^{\ast }$-algebras of type ${\mathbf{I}}_{\mathbf{2}}$

In the paper we give a characterization of a $w^{\ast }$-continuous orthogonal vector field $F$ over an $W^{\ast }$-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 $w^{\ast }$-continuous orthogonal vector field over a $W^{\ast }$-algebra of type $I_2$ is stationary.