Modular Jordan algebras of self-adjoint operators

An investigation is made into the connection between the type of a $JW$-algebra (i.e. , a weakly closed Jordan algebra of self-adjoint operators on a Hilbert space) and the type of the enveloping yon Neumann algebra. It is shown that every finite trace (faithful or normal) on a $JW$-algebra $A$ can be extended to a finite trace (faithful or normal, respectively) on the enveloping yon Neumann algebra $\mathfrak U(A)$. Using this result, it is shown that the $JW$ algebra $A$ is modular if and only if $\mathfrak U(A)$ is a finite yon Neumann algebra. If $A$ is a reversible $JW$-factor, then it has the type $\operatorname{II}_1$ if and only if $\mathfrak U(A)$ has the type $\operatorname{II}_1$.