Perturbation of a nest algebra module

1983 ◽  
Vol 93 (2) ◽  
pp. 303-306 ◽  
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.

Generators of Nest Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 565-575 ◽  
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].

On the Structure of Certain Nest Algebra Modules

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.

scholarly journals Nonlinear Isometries on Schatten-pClass in Atomic Nest Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Kan He ◽  
Qing Yuan
Keyword(s):  
Hilbert Space ◽  
Complete Characterization ◽  
Complex Hilbert Space ◽  
Nest Algebra ◽  
Nest Algebras ◽  
Surjective Isometry

LetHbe a complex Hilbert space; denote by Alg 𝒩and𝒞p(H)the atomic nest algebra associated with the atomic nest𝒩onHand the space of Schatten-pclass operators on,Hrespectively. Let𝒞p(H)∩Alg 𝒩be the space of Schatten-pclass operators in Alg 𝒩. When1≤p<+∞andp≠2, we give a complete characterization of nonlinear surjective isometries on𝒞p(H)∩Alg 𝒩. Ifp=2, we also prove that a nonlinear surjective isometry on𝒞2(H)∩Alg 𝒩is the translation of an orthogonality preserving map.

scholarly journals On Fully Stable Banach Algebra Modules and Fully Pesudo Stable Banach Algebra Modules

2018 ◽  
Vol 15 (1) ◽  
pp. 102-105 ◽  
Author(s):  
Baghdad Science Journal
Keyword(s):  
Banach Algebra ◽  
Algebra Module

The concept of fully pseudo stable Banach Algebra-module (Banach A-module) which is the generalization of fully stable Banach A-module has been introduced. In this paper we study some properties of fully stable Banach A-module and another characterization of fully pseudo stable Banach A-module has been given.

Branched Sidechain bPNA + Reports on RNA-Loop Tertiary Interactions via Triplex Hybridization Stem Replacement at Internal Domains

2019 ◽  
Author(s):  
Dennis Bong ◽  
Shiqin Miao ◽  
Yufeng Liang ◽  
Jie Mao ◽  
Ila Marathe ◽  
...  
Keyword(s):  
Peptide Nucleic Acid ◽  
Electrostatic Interactions ◽  
Noncoding Rnas ◽  
Compact Perturbation ◽  
Site Specific ◽  
New Variant ◽  
Tertiary Interactions ◽  
Structural Probes ◽  
Rna Transcript

We report herein the synthesis and characterization of bPNA+, a new variant of bifacial peptide nucleic acid (bPNA) that binds oligo T/U nucleic acids to form triplex hybrids with good affinity but half the molecular footprint of bPNA. This is accomplished via display of two melamine (M) bases melamine per lysine sidechain on bPNA+ rather than one as previously reported. Lysine derivatives bearing two bases were prepared by double reductive alkylation with melamine acetaldehyde, resulting in a tertiary amine branch point. Importantly, this amino sidechain fosters oligo T/U binding through both base-triple formation and electrostatic interactions, while maintaining selectivity and peptide solubility. The bPNA+ binding site roughly the size of a 6 base-pair stem; this relatively compact perturbation can be genetically encoded at virtually any position within an RNA transcript, and may replace existing stem elements. Subsequent triplex hybridization with fluorophore-labeled bPNA+ triplex hybrids thus accomplishes site-specific labeling of internal RNA locations without the need for chemical modification. We demonstrate herein the use of this strategy for reporting on intermolecular RNA-RNA kissing loop interactions, RNA-protein binding as well as intramolecular RNA tetraloop-tetraloop receptor binding. We anticipate that bPNA+ will have utility as structural probes for dynamic tertiary interactions in long noncoding RNAs.

Commutators in Factors of Type III

1966 ◽  
Vol 18 ◽  
pp. 1152-1160 ◽  
Author(s):  
Arlen Brown ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Complex Hilbert Space ◽  
Identity Operator ◽  
Von Neumann ◽  
Type Iii ◽  
Underlying Space ◽  
Special Cases ◽  
Weakly Closed ◽  
Algebra Of Operators

Let denote a separable, complex Hilbert space, and let R be a von Neumann algebra acting on . (A von Neumann algebra is a weakly closed, self-adjoint algebra of operators that contains the identity operator on its underlying space.) An element A of R is a commutator in R if there exist operators B and C in R such that A = BC — CB. The problem of specifying exactly which operators are commutators in R has been solved in certain special cases; e.g. if R is an algebra of type In (n < ∞) (2), and if R is a factor of type I∞ (1). It is the purpose of this note to treat the same problem in case R is a factor of type III. Our main result is the following theorem.

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

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.

A separable quasidiagonal C*-algebra with a non-quasidiagonal quotient by the compact operators

1991 ◽  
Vol 110 (1) ◽  
pp. 143-145 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Finite Rank ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
C Algebra ◽  
Alternative Characterization ◽  
Image Position ◽  
Algebra Of Operators

A C*-algebra A of operators on a separable Hilbert space H is said to be quasidiagonal if there is an increasing sequence E1, E2, … of finite-rank projections on H tending strongly to the identity and such thatas i → ∞ for T∈A. More generally a C*-algebra is quasidiagonal if there is a faithful *-representation π of A on a separable Hilbert space H such that π(A) is a quasidiagonal algebra of operators. When this is the case, there is a decomposition H = H1 ⊕ H2 ⊕ … where dim Hi < ∞ (i = 1, 2,…) such that each T∈π(A) can be written T = D + K where D= D1 ⊕ D2 ⊕ …, with Di∈L(Hi) (i = 1, 2,…), and K is a compact linear operator on H. As is well known (and readily seen), this is an alternative characterization of quasidiagonality.

scholarly journals Factorization of triangular operators and ideals through the diagonal

1997 ◽  
Vol 40 (2) ◽  
pp. 227-241 ◽  
Author(s):  
John Lindsay Orr ◽  
David R. Pitts
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Nest Algebra ◽  
Nest Algebras ◽  
Similar Criterion ◽  
Necessary And Sufficient

We give a necessary and sufficient condition to determine when an operator in the nest algebra of doubly infinite block upper triangular operators factors through a diagonal projection. An example shows that this condition does not extend to more general nest algebras, but a similar criterion yields a description of the ideals of nest algebras generated by diagonal projections.

Spectral characterization of the radical in Banach and Jordan–Banach algebras

1993 ◽  
Vol 114 (1) ◽  
pp. 31-35 ◽  
Author(s):  
Bernard Aupetit
Keyword(s):  
Representation Theory ◽  
Spectral Radius ◽  
Jacobson Radical ◽  
Classical Result ◽  
Banach Algebras ◽  
Spectral Characterization ◽  
General Situation

If a is a n × n matrix such that a + m is invertible for every invertible a + m matrix m, then a = 0, by a classical result of Perlis [8]. Unfortunately the same result is not true in general for semi-simple rings as shown by T. Laffey. In the general situation of Banach algebras, Zemánek[12] has proved that a is in the Jacobson radical of A if and only if ρ(a+x) = ρ(x), for every x in A, where ρ denotes the spectral radius. More sophisticated characterizations of the radical were given in [4] and [3], theorem 5·3·1. The arguments used in all these situations depend heavily on representation theory.

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