Abstract
The hermitian part L(A)(h) of the Banach-Lie *-algebra L(A) of multiplication operators on the W*-algebra 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*-homeornorphic to the state space of G(A). It is shown that there exists a Lie *-isomorphism phi from the quotient (A circle plus(infinity) A(OP))/K of an enveloping W*-algebra A circle plus(infinity) A(OP) of A by a weak*-closed Lie *-ideal K onto L(A), the restriction to the hermitian part ((A circle plus(infinity) A(OP))/K)(h) of which is a bi-positive real linear isometry, thereby giving a characterization of the state space of L(A). In the special case in which A is a W*-factor this leads to a further identification of the state space of L(A) in terms of the state space of A. For any W*-algebra A, the Banach-Lie *-algebra L(A) coincides with the set of generalized derivations of A, and, as an application, a formula is obtained for the norm of an element of L(A)(h) in terms of a centre -valued 'norm' on A, which is similar to that previously obtained by nori-order-theoretic methods.