Nest Algebras Research Articles

Large Perturbations of Nest Algebras

Large Perturbations of Nest Algebras

Lattices of Logmodular Algebras

A subalgebra \mathcal{A} of a C^{*} -algebra \mathcal{M} is logmodular (resp. has factorization) if the set \{a^{*}a;\ \text{$a\in\mathcal{M}$ is invertible with $a,a^{-1}\in\mathcal{A}$}\} is dense in (resp. equal to) the set of all positive and invertible elements of \mathcal{M} . In this paper, we show that the lattice of projections in a (separable) von Neumann algebra \mathcal{M} whose ranges are invariant under a logmodular algebra in \mathcal{M} , is a commutative subspace lattice. Further, if \mathcal{M} is a factor then this lattice is a nest. As a special case, it follows that all reflexive (in particular, completely distributive CSL) logmodular subalgebras of type I factors are nest algebras, thus answering in the affirmative a question by Paulsen and Raghupathi (Trans. Amer. Math. Soc. 363 (2011) 2627–2640). We also give a complete characterization of logmodular subalgebras in finite-dimensional von Neumann algebras.

Noncommutative Nest Hardy Spaces and Martingales

Abstract Let $(\mathcal{M},\tau )$ be a finite von Neumann algebra equipped with a normalized faithful trace and let $\mathbb{A}$ be a nest of order type $\mathbb{N}$. The nest algebra is defined by $H_{\infty }^{r}(\mathbb{A})=\{x\in \mathcal{M}:ex=exe,e\in \mathbb{A}\}$, and the related noncommutative Hardy space $H_{p}^{r}(\mathbb{A})$ is the completion of $H_{\infty }^{r}(\mathbb{A})$ in $L_{p}(\mathcal{M})$. In the present paper, we make a connection between $H_{p}^{r}(\mathbb{A})$ and certain noncommutative martingale Hardy space via showing that triangular truncation is a concrete martingale transform. We include several related results for noncommutative nest Hardy spaces in the spirit of recent development of noncommutative martingale theory.

Lie Modules of Banach Space Nest Algebras

In the present work, we extend to Lie modules of Banach space nest algebras a well-known characterisation of Lie ideals of (Hilbert space) nest algebras. Let A be a Banach space nest algebra and L be a weakly closed Lie A-module. We show that there exist a weakly closed A-bimodule K, a weakly closed subalgebra DK of A, and a largest weakly closed A-bimodule J contained in L,such that J⊆L⊆K+DK, with [K,A]⊆L. The first inclusion holds in general, whilst the second is shown to be valid in a class of nest algebras.

Stability analysis for the two-port networked control systems within the framework of nest algebra

Stability analysis for the two-port networked control systems within the framework of nest algebra

Lie derivable maps at unit operator on nest algebras

Let be a nontrivial nest on a separable Hilbert space and be the associated nest algebra. In this note, we prove that, if contains an atom of infinite rank or is uncountable, then the unit operator I is a Lie all-derivable point of .

2-Local Lie triple isomorphisms of nest algebras

Let be a nontrivial nest on a complex separable Hilbert space with dim , and Alg be the associated nest algebra. Suppose that is an additive surjective 2-local Lie triple isomorphism. If is not of infinite multiplicity, we prove that δ is of the form for any , where is an isomorphism or the negative of an anti-isomorphism and is a linear map with for all .

Stability analysis for linear time‐varying systems using bicoprime factorization

AbstractThis paper presents a bicoprime factorization (BCF) approach to deal with various stability problems for linear discrete time‐varying (LTV) systems within the framework of nest algebra. Based on the bicoprime factorization, we give necessary and sufficient conditions for (strongly) simultaneous stabilization of three systems, which generalize similar results for systems by coprime factorization (CF). Finally, when the bicoprime factorization for the plant and controller has additive perturbations, we derive sufficient conditions using the norm inequality for robust stability of the system.

Triangular algebras with nonlinear higher Lie n-derivation by local actions

<abstract><p>This paper was devoted to the study of the so-called nonlinear higher Lie n-derivation of triangular algebras $ \mathcal{T} $, where $ n $ is a nonnegative integer greater than two. Under some mild conditions, we proved that every nonlinear higher Lie n-derivation by local actions on the triangular algebras is of a standard form. As an application, we gave a characterization of higher Lie $ n $-derivation by local actions on upper triangular matrix algebras, block upper triangular matrix algebras and nest algebras, respectively.</p></abstract>

Morita equivalence of w*-rigged modules

The w*-rigged modules over dual operator algebras were introduced by Blecher and Kashyap as a generalization of W*-modules. In this paper, we introduce two new types of Morita equivalence between right w*-rigged modules over unital dual operator algebras and we examine whether these notions imply stable isomorphism between the corresponding modules. Furthermore, we investigate them in detail for the class of right w*-rigged modules over nest algebras, a class which was characterized by G.K. Eleftherakis.

Characterizations of Lie centralizers of triangular algebras

Let be an unital algebra over the complex field . A linear map ϕ from into itself is called a Lie centralizer at a given point if for all with ST = G. The aim of this paper is to give a description of Lie centralizers at an arbitrary but fixed point on triangular algebras. These results are then applied to nest algebras and upper triangular matrix algebras.

Nonlinear Lie Triple Higher Derivations on Triangular Algebras by Local Actions: A New Perspective

Let R be a commutative ring with unity and T be a triangular algebra over R. Let a sequence Δ={δn}n∈N of nonlinear mappings δn:T→T is a Lie triple higher derivation by local actions satisfying the equation. Under some mild conditions on T, we prove in this paper that every Lie triple higher derivation by local actions on the triangular algebras is proper. As an application, we shall give a characterization of Lie triple higher derivations by local actions on upper triangular matrix algebras and nest algebras, respectively.

C⁎-extreme maps and nests

The generalized state space SH(A) of all unital completely positive (UCP) maps on a unital C⁎-algebra A taking values in the algebra B(H) of all bounded operators on a Hilbert space H, is a C⁎-convex set. In this paper, we establish a connection between C⁎-extreme points of SH(A) and a factorization property of certain algebras associated to the UCP maps. In particular, the factorization property of some nest algebras is used to give a complete characterization of those C⁎-extreme maps which are direct sums of pure UCP maps. This significantly extends a result of Farenick and Zhou (1998) [14] from finite to infinite dimensional Hilbert spaces. Also it is shown that normal C⁎-extreme maps on type I factors are direct sums of normal pure UCP maps if and only if an associated algebra is reflexive. Further, a Krein-Milman type theorem is established for C⁎-convexity of the set SH(A) equipped with bounded weak topology, whenever A is a separable C⁎-algebra or it is a type I factor. As an application, we provide a new proof of a classical factorization result on operator-valued Hardy algebras.

Characterizing Jordan Derivable Maps on Triangular Rings by Local Actions

Suppose that is a 2‐torsion free triangular ring, and , where P is the standard idempotent of and Q = I − P. Let be a mapping (not necessarily additive) satisfying, , where A∘B = AB + BA is the Jordan product of . We obtain various equivalent conditions for δ, specifically, we show that δ is an additive derivation. Our result generalizes various results in these directions for triangular rings. As an application, δ on nest algebras are determined.

A Class of Nonlinear Nonglobal Semi-Jordan Triple Derivable Mappings on Triangular Algebras

In this paper, we proved that each nonlinear nonglobal semi-Jordan triple derivable mapping on a 2-torsion free triangular algebra is an additive derivation. As its application, we get the similar conclusion on a nest algebra or a 2-torsion free block upper triangular matrix algebra, respectively.

Generalized Jordan N-Derivations of Unital Algebras with Idempotents

Let A be a unital algebra with idempotent e over a 2-torsionfree unital commutative ring ℛ and S : A ⟶ A be an arbitrary generalized Jordan n-derivation associated with a Jordan n-derivation J . We show that, under mild conditions, every generalized Jordan n-derivation S : A ⟶ A is of the form S x = λ x + J x in the current work. As an application, we give a description of generalized Jordan derivations for the condition n = 2 on classical examples of unital algebras with idempotents: triangular algebras, matrix algebras, nest algebras, and algebras of all bounded linear operators, which generalize some known results.

Bimodules of Banach Space Nest Algebras

Abstract We extend to Banach space nest algebras the theory of essential supports and support function pairs of their bimodules, thereby obtaining Banach space counterparts of long-established results for Hilbert space nest algebras. Namely, given a Banach space nest algebra $\mathcal{A}$, we characterize the maximal and the minimal $\mathcal{A}$-bimodules having a given essential support function or support function pair. These characterizations are complete except for the minimal $\mathcal{A}$-bimodule corresponding to a support function pair, in which case we make some headway. We also show that the weakly closed bimodules of a Banach space nest algebra are exactly those that are reflexive operator spaces. To this end, we crucially prove that reflexive bimodules determine uniquely a certain class of admissible support functions.

Lie centralizers at zero products on a class of operator algebras

Let $${\mathcal {A}}$$ be an algebra. In this paper, we consider the problem of determining a linear map $$\psi $$ on $${\mathcal {A}}$$ satisfying $$a,b\in {\mathcal {A}}$$ , $$ab=0 \Longrightarrow \psi ([a,b])=[\psi (a),b] \, (C1) $$ or $$ab=0 \Longrightarrow \psi ([a,b])=[a,\psi (b)] \, (C2)$$ . We first compare linear maps satisfying (C1) or (C2), commuting linear maps, and Lie centralizers with a variety of examples. In fact, we see that linear maps satisfying (C1), (C2) and commuting linear maps are different classes of each other. Then, we introduce a class of operator algebras on Banach spaces such that if $${\mathcal {A}}$$ is in this class, then any linear map on $${\mathcal {A}}$$ satisfying (C1) (or (C2)) is a commuting linear map. As an application of these results, we characterize Lie centralizers and linear maps satisfying (C1) (or (C2)) on nest algebras.

Nest algebras in an arbitrary vector space

We examine the properties of algebras of linear transformations that leave invariant all subspaces in a totally ordered lattice of subspaces of an arbitrary vector space. We compare our results with those that apply for the corresponding algebras of bounded operators that act on a Hilbert space.

Networked Robust Stability for LTV Systems with Simultaneous Uncertainties in Plant, Controller, and Communication Channels

In this paper, we study the robust stability of a networked control system (NCS) under the framework of infinite-dimensional discrete-time linear time-varying (LTV) systems. The NCS consists of a pair of uncertain plant and controller, as well as an uncertain bilateral communication channel in between. The uncertainties in the plant and controller are measured by the gap metric. The communication channel between the plant and controller is described by a cascade of two-port networks whose transmission matrices are subject to norm-bounded additive uncertainties. Such an uncertain two-port network can model distortions and interferences occurring during control and measurement signal transmissions. The causality of the LTV subsystems is characterized by using nest algebras. A necessary and sufficient condition for the robust stability of the NCS, with the causality of all system components explicitly considered, is established in the form of an arcsine inequality which generalizes a similar result for linear time-invariant NCSs.