2
$\begingroup$

I found this theorem in Minnesota notes of Koecher [Thm. 9 p.70]:

Thm. Any semisimple Jordan algebra has a unit element.

During the text Koecher do not state that the Jordan algebra has to be finite dimensional (thus one might be thinking of general setting where infinite dimensional Jordan algebra are included). Nevertheless the proof makes use of the existence of a maximal idempotent element (which I can easily see in the finite dimensional case but I do not really see it in the infinite dimensional).

Since those are supposed to be notes (and thus might be less precise) does anyone know if the theorem holds in general or is it something that is just valid for finite dimensional Jordan algebras?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

As I was suspecting, the existence of a maximal idempotent is not assured in an infinite-dimensional simple Jordan algebra.

In fact the theorem is valid only in the finite-dimensional case. For reference see Cho-Ho Proposition 1.1.13.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .