TAILIEUCHUNG - Báo cáo toán học: "Reflexive algebras with completely distributive subspace lattices "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Reflexive lattices đại số với không gian con hoàn toàn phân phối. | J. OPERATOR THEORY 11 1984 91-108 Copyright by INCREST 1984 REFLEXIVE ALGEBRAS WITH COMPLETELY DISTRIBUTIVE SUBSPACE LATTICES ALAN HOPENWASSER CECELIA LAURIE and ROBERT MOORE The structure of a reflexive operator algebra is determined by the properties of its subspace lattice. One of the nicest properties that a lattice can possess dis-tributivity is automatic for commutative subspace lattices. Consequently CSL algebras reflexive algebras whose subspace lattices are commutative form a tractable and interesting class of operator algebras. The strongest possible distributive law permits distribution of the lattice operations over families of arbitrary cardinality. Complete lattices which satisfy this property are said to be completely distributive see Section 1 for a precise definition . Every complete lattice which is totally ordered is completely distributive on the other hand a complete Boolean algebra is completely distributive if and only if it is atomic. Thus the CSL algebras with completely distributive subspace lattices form a proper subclass of the CSL algebras which contains all nest algebras. A CSL algebra may possess many compact operators for a nest algebra the linear span of the rank one operators in the algebra is dense in the ultraweak topology or it may possess none for example a maximal abelian von Neumann algebra with no minimal projections . Complete distributivity of the subspace lattice is intimately related to the presence of compact operators in a CSL algebra. It is already known by results of Longstaff 10 11 that if a reflexive algebra has a completely distributive subspace lattice then the rank one operators in the algebra determine the subspace lattice and that if the linear span of the rank one operators in a reflexive algebra is dense in the algebra then the subspace lattice is completely distributive. In this paper we shall show that for a CSL algebra the Hilbert-Schmidt operators in the algebra are ultraweakly dense in the algebra if .

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