The Banach-Lie Algebra of Multiplication Operators on a W*-algebra

Journal Article

Hoskins, • M. Alqurashi, N. Altwaijry, C. M. Edwards, C. . 2011

• المجلة \ الصحيفة: Asian European Journal of Mathematics

مستخلص المنشور

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 from the quotient
(A 1 Aop)/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 ((A1Aop)/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.