Let $\mathcal M$ and $\mathcal N$ be two von Neumann algebras. A linear map $J:\mathcal M\to\mathcal N$ is said to be a *Jordan isomorphism* if $J$ is bijective, $*$-preserving and $J(xy+yx)=J(x)J(y)+J(y)J(x)$ for all $x,y \in \mathcal M.$

Is there any nice classification of type I von Neumann algebras up to Jordan isomorphism? Also is there is any classification of type I factors up to Jordan isomorphism?