Math. Nachr. 256, 100 – 106 (2003) / DOI 10.1002/mana.200310073 On the axiomatic definition of real JB ∗ –triples Antonio M. Peralta ∗1 1 Departamento de An´ alisis Matem´ atico, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain Received 18 December 2001, revised 19 June 2002, accepted 26 July 2002 Published online 25 June 2003 Key words Real JB*–triples, real C*–algebras, J*B–algebras MSC (2000) Primary: 17C65, 46L05, 46L70 In the last twenty years, a theory of real Jordan triples has been developed. In 1994 T. Dang and B. Russo introduced the concept of J ∗ B–triple. These J ∗ B–triples include real C ∗ –algebras and complex JB ∗ –triples. However, concerning J ∗ B–triples, an important problem was left open. Indeed, the question was whether the complexification of a J ∗ B–triple is a complex JB ∗ –triple in some norm extending the original norm. T. Dang and B. Russo solved this problem for commutative J ∗ B–triples. In this paper we characterize those J ∗ B–triples with a unitary element whose complexifications are complex JB ∗ –triples in some norm extending the original one. We actually find a necessary and sufficient new axiom to characterize those J ∗ B–triples with a unitary element which are J ∗ B–algebras in the sense of [1] or real JB ∗ –triples in the sense of [4]. 1 Introduction We recall that a real (respectively, complex) Banach Jordan triple is a real (respectively, complex) Banach space U with a continuous trilinear (respectively, bilinear in the outer variables and conjugate linear in the middle one) product U × U × U -→ U (xyz ) -→ {x, y, z } satisfying 1. {x, y, z } = {z,y,x} ; 2. Jordan Identity: L(a, b){x, y, z }-{L(a, b)x, y, z } = -{x, L(b, a)y,z } + {x, y, L(a, b)z } for all a, b, x, y, z in U , where L(x, y)z := {x, y, z }. A Banach Jordan triple U is commutative or abelian if {{x, y, z } , u, v} = {x, y, {z,u,v}} = {x, {y,z,u} ,v} for all x,y,z,u,v ∈ U . An element u ∈ U is said to be unitary if L(u, u) coincides with the identity map on U . A complex Jordan Banach triple E is said to be a (complex) JB ∗ –triple if (a) The map L(a, a) from E to E is an hermitian operator with non negative spectrum for all a in E ; (b) ‖{a, a, a}‖ = ‖a‖ 3 for all a in E . We recall that a bounded linear operator T on a complex Banach space is said to be hermitian if ‖ exp(iαT )‖ = 1 for all real α. ∗ e–mail: aperalta@ugr.es, Phone: +34 958 246311, Fax: +34 958 243272 c  2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0025-584X/03/25607-0100 $ 17.50+.50/0