6
$\begingroup$

Dye's classical theorem from 1955 states that if $\theta$ is a projection orthoisomorphism (i.e., a bijection between the projective lattices that preserves orthogonality - it then automatically preserves $0$, $1$, order, and commutativity) between two complex von Neumann algebras $M$ and $N$, and $M$ has no type $I_2$ direct summand, then $\theta$ extends to a Jordan isomorphism between $M$ and $N$.

I am interested whether the same theorem still holds for real von Neumann algebras. The answer to the question Type III factor representation by https://mathoverflow.net/users/46855/user46855 contains the claim that it holds for real $AW$*-algebras with no abelian or type $I_2$ direct summand, together with a sketch of a proof. Hence it seems to be true if we also exclude the abelian summand. But I cannot find a reference to a complete proof of this result.

So my question is as follows. Does anyone have a reference to the following result: if $\theta$ is a projection orthoisomorphism between real von Neumann algebras $M$ and $N$, with $M$ having no abelian or type $I_2$ direct summand, does $\theta$ extend to a Jordan isomorphism between $M$ and $N$?

$\endgroup$
1
$\begingroup$

Corollary 2 in the paper "On Dye's Theorem for Jordan operator algebras'' is Dye's Theorem for $JW$-algebras $M$ and $N$ and $M$ has no type $I_2$ direct summand. The result now follows since the selfadjoint part of a real von Neumann algebra is a $JW$-algebra.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.