The Banach-Lie algebra of multiplication operators on a JBW^*-triple

The Banach-Lie algebra L (A) of multiplication operators on the JB*-triple A is introduced and it is shown that the hermitian part L (A)h of L (A) is a unital GM-space the base of the dual cone in the dual GL-space (L (A)h)* of which is affine isomorphic and weak*-homeomorphic to the state space of L (A). In the case in which A is a JBW*-triple, it is shown that tripotents u and v in A are orthogonal if and only if the corresponding multiplication operators in the unital GM-space L (A)h satisfy0 ≤ D (u, u) + D (v, v) ≤ idA, and that u is a pre-associate of v if and only ifD (u, u) ≤ D (v, v) . © 2008 Elsevier Inc. All rights reserved.