Narrow Results

Given a W*-algebra with a W*-dynamics τ, we prove the existence of the perturbed W*-dynamics for a large class of unbounded perturbations. We compute its Liouvillean. If τ has a β-KMS state, and the perturbation satisfies some mild assumptions related to the Golden–Thompson inequality, we prove the existence of a β-KMS state for the perturbed W*-dynamics. These results extend the well known constructions due to Araki valid for bounded perturbations.

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 ⊕_∞ A^op)/K of an enveloping W*-algebra A ⊕_∞ A^op of A by a weak*-closed Lie *-ideal K onto , 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 . 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.