THE BANACH-LIE *-ALGEBRA OF MULTIPLICATION OPERATORS ON A W*-ALGEBRA
Abstract
The hermitian part 
of the Banach-Lie *-algebra 
of multiplication operators on the W*-algebra A is a unital GM-space, the base of the dual cone in the dual GL-space 
of which is affine isomorphic and weak*-homeomorphic to the state space of 
. It is shown that there exists a Lie *-isomorphism ϕ from the quotient (A ⊕∞ Aop)/K of an enveloping W*-algebra A ⊕∞ Aop of A by a weak*-closed Lie *-ideal K onto 
, the restriction to the hermitian part ((A ⊕∞ Aop)/K)h of which is a bi-positive real linear isometry, thereby giving a characterization of the state space of 
. In the special case in which A is a W*-factor this leads to a further identification of the state space of 
in terms of the state space of A. For any W*-algebra A, the Banach-Lie *-algebra 
coincides with the set of generalized derivations of A, and, as an application, a formula is obtained for the norm of an element of 
in terms of a centre-valued 'norm' on A, which is similar to that previously obtained by non-order-theoretic methods.
