Real Banach Jordan triples

Abstract

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 J B ∗ J{B^ \ast } -triple in arbitrary dimensions. These spaces include real C ∗ {C^ \ast } -algebras and J B ∗ J{B^\ast } -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 J B ∗ J{B^\ast } -triples (see §2), we can now propose a new definition of a real J B ∗ J{B^\ast } -triple, which we call a J ∗ B {J^\ast }B -triple. Our J ∗ B {J^\ast }B -triples include real C ∗ {C^\ast } -algebras and complex J B ∗ J{B^\ast } -triples. Our main theorem is a structure theorem of Gelfand-Naimark type for commutative J ∗ B {J^\ast }B -triples.