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In 1934, Jordan, von Neumann and Wigner gave a nice classification of finite-dimensional simple Jordan algebras that are 'formally real', meaning that a sum of squares is zero only if each term in the sum is zero. In 1983 Zelmanov generalized this to all simple Jordan algebras, but unlike the original result, Zelmanov's 'classification' is not a neat list. What about formally real Jordan algebras, not necessarily finite-dimensional? What does the classification of these look like?

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Thanks for the welcome and help... –  John Baez Nov 11 '10 at 5:59
    
The work by Zelmanov that you refer to appears to be this one: ams.org/mathscinet-getitem?mr=688595 (MR0688595, for future reference). –  Harald Hanche-Olsen Nov 11 '10 at 14:27
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The formally real Jordan algebras include the class of JB-algebras, a class of normed Jordan algebra which is the Jordan algebra equivalent of C*-algebras. From this you might imagine that obtaining a complete classification is a rather non-trivial task. In this class you also find JW-algebras, which are JB-algebras with a predual, thus corresponding to von Neumann algebras. Some of the concepts from C*- and von Neumann algebra theory carry over to the Jordan algebra setting, but this margin is too narrow to summarize what is known. With apologies for tooting my own horn here, in 1984 I coauthored a book “Jordan Operator Algebras” with Erling Størmer which pretty much summarized the state of the art at the time. (Since then I have left that field, so I don't know if a lot has happened since.)

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JB, were you just fishing for someone to mention JB-algebras? –  Allen Knutson Nov 9 '10 at 13:42
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Heh. Well, JB in this context stands for Jordan–Banach. –  Harald Hanche-Olsen Nov 9 '10 at 15:09
    
Thanks, Harald. No thanks, Allen. :-) I would really like to see a wild multitude of simple formally real Jordan algebras. I'm not sure I see them yet. Harald, I guess you're hinting that I can get such a thing from any C*-algebra that lacks nontrivial star-ideals. But this reminds me that my question said nothing about a norm, or topology. I know lots of infinite-dimensional C*-algebras that lack nontrivial closed star-ideals, but how about C*-algebras that don't have any nontrivial star-ideals? –  John Baez Nov 11 '10 at 5:58
    
John, I was painfully aware at the time that I did not really answer your question. Still, I hoped (and still hope) that at least it did shed some light on some corner of it. Regarding ideals, I am sure you are aware that in a unital C*-algebra (or unital Banach algebra more generally), the closure of a nontrivial ideal is again nontrivial, since a neighbourhood of the unit consists of invertible elements. Non-unital algebras is presumably a whole different kettle of fish. –  Harald Hanche-Olsen Nov 11 '10 at 14:13
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