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Title:  The Banach Lie algebra of multiplication operators on a von Neumann algebra 
Authors:  Al Qurashi, M. Al Twaijry, N. .M. Edwards, C. Hoskin, C. 
Keywords:  von Neumann algebra, Banach Lie algebra 
Issue Date:  2011 
Publisher:  AsianEuropean Journal of Mathematics (AEJM) 
Abstract:  The hermitian part L(A)h of the BanachLie _algebra L(A) of
multiplication operators on the W_algebra A is a unital GMspace, the base
of the dual cone in the dual GLspace (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 bipositive
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 centrevalued `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 nonorder theoretic methods. 
URI:  http://hdl.handle.net/123456789/18133 
Appears in Collections:  College of Science

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