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

摘要

The Banach-Lie algebra C(A) of multiplication operators on the JB*-triple A is introduced and it is shown that the hermitian part C(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 nu in A are orthogonal if and only if the corresponding multiplication operators in the unital GM-space L(A)(h) satisfy 0 <= D(u, u) + D(nu, nu) <= id(A), and tha u is pre-associate of nu if add only if D(u, u) <= D(nu, nu). (c) 2008 Elsevier Inc. All rights reserved.