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

The Banach–Lie algebra L(A)L(A) of multiplication operators on the JB∗-triple A   is introduced and it is shown that the hermitian part L(A)hL(A)h of L(A)L(A) is a unital GM-space the base of the dual cone in the dual GL-space (L(A)h)∗(L(A)h)∗ of which is affine isomorphic and weak∗-homeomorphic to the state space of L(A)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)hL(A)h satisfy0⩽D(u,u)+D(v,v)⩽idA,0⩽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).D(u,u)⩽D(v,v).