Weakly closed Jordan ideals in nest algebras on Banach spaces

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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Perturbation of a nest algebra module

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410006059x ◽

1983 ◽

Vol 93 (2) ◽

pp. 303-306 ◽

Cited By ~ 1

Author(s):

Sotirios Karanasios

Keyword(s):

General Situation ◽

Compact Perturbation ◽

Nest Algebra ◽

Nest Algebras ◽

Weakly Closed ◽

Carry Over ◽

Algebra Module ◽

Algebra Of Operators

Fall, Arveson and Muhly(4) characterized the compact perturbation of nest algebras. In fact they proved that the compact perturbation of a nest algebra corresponding to a nest of projections is the algebra of operators which are quasitriangular relative to this nest. Erdos and Power(3) investigated weakly closed ideals and modules of nest algebras and these exhibit properties that are very close to the properties of the nest algebras themselves. They also showed that in certain cases, as in the case when the homomorphism which determines the nest algebra module is continuous, the results of Fall, Arveson and Muhly carry over to the more general situation. In this paper we provide a characterization of the compact perturbation of any nest algebra module.

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Lie triple derivations of nest algebras on Banach spaces

Linear Algebra and its Applications ◽

10.1016/j.laa.2011.12.018 ◽

2012 ◽

Vol 436 (9) ◽

pp. 3443-3462 ◽

Cited By ~ 5

Author(s):

Shanli Sun ◽

Xuefeng Ma

Keyword(s):

Banach Spaces ◽

Nest Algebras

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Characterizing derivations for any nest algebras on Banach spaces by their behaviors at an injective operator

Linear Algebra and its Applications ◽

10.1016/j.laa.2014.02.019 ◽

2014 ◽

Vol 449 ◽

pp. 312-333 ◽

Cited By ~ 6

Author(s):

Yanfang Zhang ◽

Jinchuan Hou ◽

Xiaofei Qi

Keyword(s):

Banach Spaces ◽

Nest Algebras ◽

Injective Operator

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LIE-TYPE DERIVATIONS OF NEST ALGEBRAS ON BANACH SPACES AND RELATED TOPICS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788721000148 ◽

2021 ◽

pp. 1-40

Author(s):

FENG WEI ◽

YUHAO ZHANG

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Linear Mapping ◽

Linear Operators ◽

Future Research ◽

Complex Field ◽

Nest Algebra ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Nest Algebras

Abstract Let $\mathcal {X}$ be a Banach space over the complex field $\mathbb {C}$ and $\mathcal {B(X)}$ be the algebra of all bounded linear operators on $\mathcal {X}$ . Let $\mathcal {N}$ be a nontrivial nest on $\mathcal {X}$ , $\text {Alg}\mathcal {N}$ be the nest algebra associated with $\mathcal {N}$ , and $L\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ be a linear mapping. Suppose that $p_n(x_1,x_2,\ldots ,x_n)$ is an $(n-1)\,$ th commutator defined by n indeterminates $x_1, x_2, \ldots , x_n$ . It is shown that L satisfies the rule $$ \begin{align*}L(p_n(A_1, A_2, \ldots, A_n))=\sum_{k=1}^{n}p_n(A_1, \ldots, A_{k-1}, L(A_k), A_{k+1}, \ldots, A_n) \end{align*} $$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ if and only if there exist a linear derivation $D\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ and a linear mapping $H\colon \text {Alg}\mathcal {N}\longrightarrow \mathbb {C}I$ vanishing on each $(n-1)\,$ th commutator $p_n(A_1,A_2,\ldots , A_n)$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ such that $L(A)=D(A)+H(A)$ for all $A\in \text {Alg}\mathcal {N}$ . We also propose some related topics for future research.

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Nuclearity and Banach spaces

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500026274 ◽

1977 ◽

Vol 20 (3) ◽

pp. 205-209 ◽

Cited By ~ 8

Author(s):

Manuel Valdivia

Keyword(s):

Banach Spaces ◽

Weak Topology ◽

Fundamental System ◽

Nuclear Space ◽

Infinite Dimensional ◽

Separable Predual ◽

Weakly Closed ◽

The One

SummaryLet E be a nuclear space provided with a topology different from the weak topology. Let {Ai: i ∈ I} be a fundamental system of equicontinuous subsets of the topological dual E' of E. If {Fi: i ∈ I} is a family of infinite dimensional Banach spaces with separable predual, there is a fundamental system {Bi: i ∈ I} of weakly closed absolutely convex equicontinuous subsets of E'such that is norm-isomorphic to Fi, for each i ∈ I. Other results related with the one above are also given.

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