On the structure of weakly closed ternary rings of
                    operators

Several types of circular DNA molecules are now known. These are classified as single-stranded rings, covalently closed duplex rings, and weakly bonded duplex rings containing an interruption in one or both strands. Single rings are exemplified by the viral DNA from ϕX174 bacteriophage. Duplex rings appear to exist in a twisted configuration in neutral salt solutions at room temperature. Examples of such molecules are the DNA's from the papova group of tumor viruses and certain intracellular forms of ϕX and λ-DNA. These DNA's have several common properties which derive from the topological requirement that the winding number in such molecules is invariant. They sediment abnormally rapidly in alkaline (denaturing) solvents because of the topological barrier to unwinding. For the same basic reason these DNA's are thermodynamically more stable than the strand separable DNA's in thermal and alkaline melting experiments. The introduction of one single strand scission has a profound effect on the properties of closed circular duplex DNA's. In neutral solutions a scission appears to generate a swivel in the complementary strand at a site in the helix opposite to the scission. The twists are then released and a slower sedimenting, weakly closed circular duplex is formed. Such circular duplexes exhibit normal melting behavior, and in alkali dissociate to form circular and linear single strands which sediment at different velocities. Weakly closed circular duplexes containing an interruption in each strand are formed by intramolecular cyclization of viral λ-DNA. A third kind of weakly closed circular duplex is formed by reannealing single strands derived from circularly permuted T2 DNA. These reconstituted duplexes again contain an interruption in each strand though not necessarily regularly spaced with respect to each other.

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On the Structure of Certain Nest Algebra Modules

Canadian Journal of Mathematics ◽

10.4153/cjm-1987-065-7 ◽

1987 ◽

Vol 39 (6) ◽

pp. 1405-1412

Author(s):

G. J. Knowles

Keyword(s):

Operator Algebras ◽

Adjoint Operator ◽

Systematic Investigation ◽

Unique Determination ◽

Nest Algebra ◽

Minimal Elements ◽

Open Questions ◽

Weakly Closed ◽

Order Homomorphisms

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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Generators of Nest Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1974-052-8 ◽

1974 ◽

Vol 26 (3) ◽

pp. 565-575 ◽

Cited By ~ 3

Author(s):

W. E. Longstaff

Keyword(s):

Hilbert Space ◽

Separable Hilbert Space ◽

Linear Operators ◽

Nest Algebra ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Closed Algebra ◽

Nest Algebras ◽

Weakly Closed

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

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Completions of Boolean algebras of projections and weak-star closures of C*-algebras on dual Banach spaces

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091509000406 ◽

2011 ◽

Vol 54 (2) ◽

pp. 515-529

Author(s):

Philip G. Spain

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Boolean Algebras ◽

C Algebra ◽

C Algebras ◽

Weak Star ◽

Weakly Closed ◽

Operator Closure ◽

Dual Banach Space ◽

Completion Theorem

AbstractPalmer has shown that those hermitians in the weak-star operator closure of a commutative C*-algebra represented on a dual Banach space X that are known to commute with the initial C*-algebra form the real part of a weakly closed C*-algebra on X. Relying on a result of Murphy, it is shown in this paper that this last proviso may be dropped, and that the weak-star closure is even a W*-algebra.When the dual Banach space X is separable, one can prove a similar result for C*-equivalent algebras, via a ‘separable patch’ completion theorem for Boolean algebras of projections on such spaces.

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The Slice Map Problem for σ-Weakly Closed Subspaces of Von Neumann Algebras

Transactions of the American Mathematical Society ◽

10.2307/1999389 ◽

1983 ◽

Vol 279 (1) ◽

pp. 357 ◽

Cited By ~ 1

Author(s):

Jon Kraus

Keyword(s):

Von Neumann Algebras ◽

Von Neumann ◽

Weakly Closed

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On lattices of z-ideals of function rings

Mathematica Slovaca ◽

10.1515/ms-2017-0099 ◽

2018 ◽

Vol 68 (2) ◽

pp. 271-284 ◽

Cited By ~ 2

Author(s):

Themba Dube ◽

Oghenetega Ighedo

Keyword(s):

The Other ◽

Regularity Conditions ◽

Other Hand ◽

Maximal Ideals ◽

Algebraic Properties ◽

Weakly Closed ◽

Frame Homomorphism

Abstract An ideal I of a ring A is a z-ideal if whenever a, b ∈ A belong to the same maximal ideals of A and a ∈ I, then b ∈ I as well. On the other hand, an ideal J of A is a d-ideal if Ann2(a) ⊆ J for every a ∈ J. It is known that the lattices Z(L) and D(L) of the ring 𝓡L of continuous real-valued functions on a frame L, consisting of z-ideals and d-ideals of 𝓡L, respectively, are coherent frames. In this paper we characterize, in terms of the frame-theoretic properties of L (and, in some cases, the algebraic properties of the ring 𝓡L), those L for which Z(L) and D(L) satisfy the various regularity conditions on algebraic frames introduced by Martínez and Zenk [20]. Every frame homomorphism h : L → M induces a coherent map Z(h) : Z(L) → Z(M). Conditions are given of when this map is closed, or weakly closed in the sense Martínez [19]. The case of openness of this map was discussed in [11]. We also prove that, as in the case of the ring C(X), the sum of two z-ideals of 𝓡L is a z-ideal.

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