UC Irvine

A theory of real Jordan triples and real bounded symmetric domains in finite dimensions was developed by Loos. Upmeier has proposed a definition of a real triple in arbitrary dimensions. These spaces include real Calgebras and JB*-triples considered as vector spaces over the reals and have the property that their open unit balls are real bounded symmetric domains. This, together with the observation that many of the more recent techniques in Jordan theory rely on functional analysis and algebra rather than holomorphy, suggests that it may be possible to develop a real theory and to explore its relationship with the complex theory. In this paper we employ a Banach algebraic approach to real Banach Jordan triples. Because of our recent observation on commutative triples we can now propose a new definition of a real triple, which we call a triple. Our triples include real C*-algebras and complex JB*- triples. Our main theorem is a structure theorem of Gelfand-Naimark type for commutative c*-triples. © 1994 American Mathematical Society.