The Banach Lie algebra of multiplication operators on a von Neumann algebra

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 a_ne isomorphic and weak_-homeomorphic to the state space of L(A). It is shown that there exists a Lie _-isomorphism _ from the quotient (A_1Aop)=K of an enveloping W_-algebra A _1 Aop of A by a weak_-closed Lie _-ideal K onto L(A), the restriction to the hermitian part ((A _1 Aop)=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 identi_cation of the state space of L(A) in terms of the state space of A. 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. For aW_-algebra A, L(A) coincides with the set of generalized derivations of A, and a similar formula was previously obtained by non-order theoretic methods.