σ-Weakly Closed Modules of Certain Reflexive Operator Algebras

Let be a nest algebra of operators on some Hilbert space H. Weakly closed -modules were first studied by J. Erdos and S. Power in [4]. It became apparent that many interesting classes of non self-adjoint operator algebras arise as just such a module. This paper undertakes a systematic investigation of the correspondence which arises between such modules and order homomorphisms from Lat into itself. This perspective provides a basis to answer some open questions arising from [4]. In particular, the questions concerning unique “determination” and characterization of maximal and minimal elements under this correspondence, are resolved. This is then used to establish when the determining homomorphism is unique.

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On Operator Algebras and Invariant Subspaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-129-9 ◽

1969 ◽

Vol 21 ◽

pp. 1178-1181 ◽

Cited By ~ 10

Author(s):

Chandler Davis ◽

Heydar Radjavi ◽

Peter Rosenthal

Keyword(s):

Hilbert Space ◽

Elementary Proof ◽

Operator Algebras ◽

Equivalent Form ◽

Invariant Subspaces ◽

Complex Hilbert Space ◽

Closed Algebra ◽

Weakly Closed ◽

Special Case

If is a collection of operators on the complex Hilbert space , then the lattice of all subspaces of which are invariant under every operator in is denoted by Lat . An algebra of operators on is defined (3; 4) to be reflexive if for every operator B on the inclusion Lat ⊆ Lat B implies .Arveson (1) has proved the following theorem. (The abbreviation “m.a.s.a.” stands for “maximal abelian self-adjoint algebra”.)ARVESON's THEOREM. Ifis a weakly closed algebra which contains an m.a.s.a.y and if Lat, then is the algebra of all operators on .A generalization of Arveson's Theorem was given in (3). Another generalization is Theorem 2 below, an equivalent form of which is Corollary 3. This theorem was motivated by the following very elementary proof of a special case of Arveson's Theorem.

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Some properties of prereflexive subspaces of operators

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171298000787 ◽

1998 ◽

Vol 21 (3) ◽

pp. 565-570

Author(s):

Jiankui Li

Keyword(s):

Closed Subspace ◽

Weakly Closed ◽

Reflexive Operator ◽

Equivalent Conditions

In the paper, we define a notion of prereflexivity for subspaces, give several equivalent conditions of this notion and prove that ifS⫅L(H)is prereflexive, then everyσ-weakly closed subspace ofSis prereflexive if and only ifShas the property WP(see definition 2.11). By our result, we construct a reflexive operatorAsuch thatA⊕0is not prereflexive.

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