A simple Euclidean Jordan algebra (i.e. a factor) is either a spin-factor, the matrices over the reals/complex-numbers/quaternions or the exceptional Albert algebra of 3x3 octonian matrices.

My question is the following: Is it true that for any atomic idempotents $p$ and $q$ in a factor, there exists an order isomorphism $\Phi$ such that $\Phi(p)=q$ and $\Phi(q)=p$.

An atomic idempotent in the first cases can all be seen as coming from the underlying vector space. I.e. it is $\lvert v\rangle\langle v\rvert$, and this statement then becomes equivalent to asking whether there is a 'unitary' that interchanges two arbitrary normalised vectors - which is true.

So the only factor left to check is the Albert algebra. Is it possible to do the same sort of logic as above?


This is true. An abstract proof of this for any JBW-algebra factor is given in Jordan Operator Algebras by Hanche-Olsen and Størmer, proposition 5.3.2.


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