Formally real Jordan algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:56:34Z http://mathoverflow.net/feeds/question/45410 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45410/formally-real-jordan-algebras Formally real Jordan algebras John Baez 2010-11-09T06:24:36Z 2010-11-09T09:54:47Z <p>In 1934, Jordan, von Neumann and Wigner gave a <a href="http://en.wikipedia.org/wiki/Jordan_algebra#Formally_real_Jordan_algebras" rel="nofollow">nice classification</a> 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 <em>all</em> 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?</p> http://mathoverflow.net/questions/45410/formally-real-jordan-algebras/45417#45417 Answer by Harald Hanche-Olsen for Formally real Jordan algebras Harald Hanche-Olsen 2010-11-09T09:54:47Z 2010-11-09T09:54:47Z <p>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 “<a href="http://www.math.ntnu.no/~hanche/joa/" rel="nofollow">Jordan Operator Algebras</a>” 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.)</p>