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 satisfy 0 D(u, u) +D(v, v) idA, and that u is a pre-associate of v if and only if D(u, u) D(v, v).