The Banach-Lie*-algebra of multiplication operators on a W*-algebra

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*-homeomorphic to the state space of L(A). It is shown that there exists a Lie *-isomorphism φ from the quotient (A ⊕ ∞ A op)/K of an enveloping W*-algebra A ⊕ ∞ A op of A by a weak*-closed Lie *-ideal K onto L(A), the restriction to the hermitian part ((A ⊕ ∞ 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 non-order-theoretic methods. © 2011 World Scientific Publishing Company.