Nest Algebras Research Articles

Generalized interpolation in finite maximal subdiagonal algebras

Non-selfadjoint operator algebras have been studied since the paper of Kadison and Singer in 1960. In [1], Arveson introduced the notion of subdiagonal algebras as the generalization of weak *-Dirichlet algebras and studied the analyticity of operator algebras. After that, we have many papers about non-selfadjoint algebras in this direction: nest algebras, CSL algebras, reflexive algebras, analytic operator algebras, analytic crossed products and so on. Since the notion of subdiagonal algebras is the analogue of weak *-Dirichlet algebras, subdiagonal algebras have many fruitful properties from the theory of function algebras. Thus, we have several attempts in this direction: Beurling–Lax–Halmos theorem for invariant subspaces, maximality, factorization theorem and so on.

Local derivation of nest algebras

We show that every strongly continuous local derivation on a nest algebra is a derivation.

Local derivations of nest algebras

Let X be an arbitrary reflexive Banach space, and let N \mathcal {N} be a nest on X. Denote by D ( N ) \mathcal {D}(\mathcal {N}) the set of all derivations from Alg ⁡ N \operatorname {Alg}\mathcal {N} into Alg ⁡ N \operatorname {Alg}\mathcal {N} . For N ⊂ N N \subset \mathcal {N} , we set N − = ∨ { M ∈ N : M ⊂ N } {N_ - } = \vee \{ M \in \mathcal {N}:M \subset N\} . We also write 0 − = 0 {0_ - } = 0 . Finally, for E , F ∈ N E, F \in \mathcal {N} define ( E , F ] = { K ∈ N : E ⊂ K ⊆ F } (E,F] = \{ K \in \mathcal {N}:E \subset K \subseteq F\} . In this paper we prove that a sufficient condition for D ( N ) \mathcal {D}(\mathcal {N}) to be (topologically) algebraically reflexive is that for all 0 ≠ E ∈ N 0 \ne E \in \mathcal {N} and for all X ≠ F ∈ N X \ne F \in \mathcal {N} , there exist M ∈ ( 0 , E ] M \in (0,E] and N ∈ ( F , X ] N \in (F,X] , such that M − ⊂ M {M_ - } \subset M and N − ⊂ N {N_ - } \subset N . In particular, we prove that this condition automatically holds for nests acting on finite-dimensional Banach spaces.

Spectral and Fredholm properties of operators in elementary nest algebras

Some spectral and Fredholm properties are proved for linear operators which leave invariant certain nests of closed subspaces.

Triangular algebras and ideals of nest algebras

Triangular algebras and ideals of nest algebras

On some ideals of nest algebras

The purpose of this paper is to characterize the relations of some ideals of a nest algebra such as the Jacobson radical R N {R_\mathcal {N}} , Larson strong radical R N ∞ R_\mathcal {N}^\infty , and ideal G \mathcal {G} , and to describe all the ideals between R N {R_\mathcal {N}} and G \mathcal {G} .

Close CSL algebras are similar

In 1972 Kadison and Kastler [12] asked whether two close von Neumann subalgebras of ~ ( J f ) are unitarily equivalent by a unitary close to the identity. This question was answered affirmatively for certain classes of von Neumann algebras in [5, 4, 6, 15]. The nonselfadjoint case was first considered by Lance who showed in [13] that close nest algebras are similar via a similarity which is close to the identity. For nest algebras, there is a close connection between the question of whether closeness of algebras implies similarity and the classification of nest algebras up to similarity: this connection may be seen in the papers [7] and [14] and we shall discuss it further in Remark 8 below. The desire to extend the classification of nest algebras to the class of reflexive algebras with commutative subspace lattice was one of the motivations for the work on perturbations of matrix algebras begun in [3] and continuing with perturbations of suboolean operator algebras in [8]. In [16], we proved that if ~ ' is a hyperreflexive CSL algebra whose lattice is atomic and satisfies a certain technical condition, then any other CSL algebra whose unit ball is sufficiently close to the unit ball of d has the property that ~ is similar to d . This result was extended in [9] to the class of all hyperreflexive CSL algebras whose lattice is completely distributive. The purpose of the present paper is to remove the hypothesis of hyperreflexivity and complete distributivity from the results described above. We show that the class of all CSL algebras is stable under small perturbations. In particular, we prove Theorem 6 below, which gives a complete answer to Problem 5 of the section entitled "Open Problems" of [11].

Perturbations of certain reflexive algebras

In this note we use cohomological techniques to prove that if there is a linear map between two CSL algebras which is to the identity, then the two CSL algebras are similar. We use our result to show that if 2f is a purely atomic, hyperreflexive CSL with uniform infinite multiplicity which satisfies the 4-cycle interpolation condition, then there are constants δ, C > 0 such that whenever Jt is another CSL such that d(Algi?, MgJt) < δ , then there is an invertible operator S such that S k\gS?S~ι = Alg^ and ||S|| US1 II < 1 4- Cd(Alg&, 1. Introduction and preliminaries. In this paper, we consider two related types of perturbation questions for CSL algebras. The first deals with the problem of perturbing a linear isomorphism between two algebras to obtain an algebra isomorphism, and the second deals with the problem of deciding whether, for two such algebras, closeness implies isomorphism. Perturbation questions of this kind were considered by Kadison and Kastler for von Neumann algebras in [18] and further studied by many authors, including Christensen ([3, 4, 5]). Johnson ([16]) and Raeburn and Taylor ([21]) obtained results concerning perturbations of closed subalgebras of Banach algebras. Their results show that if $/ is a closed subalgebra of a Banach algebra 3S and certain cohomology groups for sf vanish, then any closed subalgebra of 3S close to sf is actually isomorphic to stf . The nonself adjoint case was considered first for nest algebras by Lance in [19]. Perturbations of other nonself adjoint operator algebras were considered by Choi and Davidson ([2]), Davidson ([6]). In §2, we prove Theorem 3 which shows that if two CSL algebras s/ and J^2 are linearly isomorphic via an isomorphism to the identity, they they are actually spatially isomorphic via an isomorphism which is to a unitary equivalence. In §3, we introduce the 4-cycle interpolation property, which is closely related to a lattice condition appearing in [12] and to the notion of interpolating lattice introduced in [7]. The main result of §3, Theorem 16, shows that if sfγ is a CSL algebra which is sufficiently to a purely atomic,

The Maximal Ideals of a Nest Algebra

The Maximal Ideals of a Nest Algebra

Formula omitted]∞ design of general multirate sampled-data control systems

Direct digital design of general multirate sampled-data systems is considered. To tackle causality constraints, a new and natural framework is proposed using nest operators and nest algebras. Based on this framework explicit solutions to the H ∞ and H 2 multirate control problems are developed in the frequency domain.

The Jacobson radical of a CSL algebra

Extrapolating from Ringrose's characterization of the Jacobson radical of a nest algebra, Hopenwasser conjectured that the radical of a CSL algebra coincides with the Ringrose ideal (the closure of the union of zero diagonal elements with respect to finite sublattices). A general interpolation theorem is proved that reduces this conjecture for completely distributive lattices to a strictly combinatorial problem. This problem is solved for all width two lattices (with no restriction of complete distributivity), verifying the conjecture in this case. In [9] (cf. [3, Chapter 6]), Ringrose characterizes the Jacobson radical of a nest algebra. In [6], Hopenwasser described the appropriate generalization for reflexive algebras with commutative subspace lattice (CSL algebras), and verified his conjecture in a few special cases. The analogy with the nest case was pushed further in [8], clarifying the role of the carrier space. In this paper, we verify this conjecture for all width two CSL algebras. Moreover, we establish a framework for attacking the general problem and make significant progress in this direction. Recall that the Jacobson radical of a Banach algebra is the intersection of the kernels of all irreducible continuous representations. It also coincides with those elements T such that AT is quasinilpotent for every A in the algebra. Thus when 9 is a finite CSL, Alg(SF) is a finite matrix algebra with certain coefficients running over W(X) (and the rest 0). There is a unique contractive projection A, of the algebra onto the diagonal = Alg(S,) nAlg(F)* given by A_(T) = ZE ETE as E runs over the atoms of S . It is easy to see that the radical of Alg(Sr) is precisely the set of zero diagonal elements (T such that A_(T) = 0), which we denote Algo(SF). Indeed, this is a nilpotent ideal of order at most 1 I. Furthermore, it is easy to verify that Algo(F) n Alg(2') is an ideal in Alg(2') for all finite sublattices F of 2'; and indeed, it is a nilpotent ideal. Thus it is contained in the radical. It follows that the norm closure

Jacobson Radicals of Nest Algebras in Factors

Definition. Let $\beta$ be a nest in a separably acting type ${\text {I}}{{\text {I}}_\infty }$ factor $\mathcal {M}$. An element $P \in \beta \backslash \{ 0,I\}$ is said to be a singular point of $\beta$ if it satisfies either of the following conditions: (1) There is a strictly increasing sequence $\{ {Q_n}\} \subseteq \beta ,\;{\lim _{n \to \infty }}{Q_n} = P$, and $P - {Q_n}$ is infinite for each $n \in \mathbb {N}$. Also, there is a projection $Q \in \beta$ such that $Q > P$ and $Q - P$ is finite. (2) There is a strictly decreasing sequence $\{ {Q_n}\} \subseteq \beta ,\;{\lim _{n \to \infty }}{Q_n} = P$, and ${Q_n} - P$ is infinite for each $n \in \mathbb {N}$. Also, there is a projection $Q \in \beta$ such that $Q < P$ and $P - Q$ is finite. Main Theorem. Let $\beta$ be a nest in a separably acting factor $\mathcal {M}$. (1) If $\mathcal {M}$ is of type ${\text {I}}{{\text {I}}_\infty }$, then a necessary and sufficient condition for the Jacobson radical ${\mathcal {R}_\beta }$ of $\operatorname {alg} \beta$ to be a norm-closed singly generated ideal of $\operatorname {alg} \beta$ is that the nest $\beta$ is countable and it does not contain a singular point. (2) If $\mathcal {M}$ is of type ${\text {I}}{{\text {I}}_1}$ or type ${\text {III}}$, then a necessary and sufficient condition for the Jacobson radical ${\mathcal {R}_\beta }$ of $\operatorname {alg} \beta$ to be a norm-closed singly generated ideal of $\operatorname {alg} \beta$ is that the nest $\beta$ is countable. (3) In (1) and (2) the single generation is equivalent to countable generation.

Additive Derivations of Nest Algebras

Additive Derivations of Nest Algebras

Stabilizability and Existence of System Representations for Discrete-Time Time-Varying Systems

In this paper, right and left representations as an alternate, but equivalent, framework to coprime factorizations of operators are developed. The main theorem of the paper establishes that a linear, time-varying, discrete-time plant is stabilizable if and only if its graph can be represented as the range (respectively, kernel) of a causal, bounded operator which is left (respectively, right) invertible. The proof relies on certain factorization theorems of Arveson for nest algebras. The paper extends the Youla parametrization of all stabilizing compensators to this framework. Also, it is proven that a time-invariant plant that is not stabilizable by a time-invariant compensator is not stabilizable with a time-varying compensator. An example of a time-varying plant of Feintuch is considered and shown to be not stabilizable. Finally, the continuous-time case is examined and the problems encountered in extending the proof are discussed. However, it is shown that a stabilizable plant must have a closed graph and this is used to prove that an example of a time-invariant, continuous-time system of Shefi is not stabilizable.

Analytic Taf Algebras

AbstractA strongly maximal triangular AF algebra which is defined by a realvalued cocycle is said to be analytic. Formulas for generic cocycles are given separately for both the integer-valued case and the real-valued coboundary case, and also for certain nest algebras. In the case of an integer-valued cocycle, there is an associated partial homeomorphismof the maximal ideal space of the diagonal. If the partial homeomorphism extends to a homeomorphism, then the algebra embeds in a crossed product. This occurs for a large class of subalgebras of UHF algebras, but an example shows that this does not always occur. An example is given of a triangular AF algebra which is analytic via a coboundary but is not a nest algebra; also, it is shown that a nest algebra need not be analytic

The First Cohomology Groups of Nest Algebras on Normed Spaces

In this short note we prove that all the first cohomology groups of nest algebras acting on a normed space are trivial when the coefficients lie in any weakly closed bimodule which contains the nest algebra

Nest Representations and Dynamical Systems

A nest representation is a generalized irreducible representation of a Banach algebra on Hubert space. For a free, discrete dynamical system, every ideal in the semicrossed product algebra is equal to an intersection of kernels of nest representations; a proper subset of all such kernels is a topological space homeomorphic to the space of all (discrete) finite arcs in the dynamical system, with the subarc topology. For certain continuous dynamical systems, the corresponding set of kernels is isomorphic to the topological space of (continuous) finite arcs. A Banach algebra isomorphism of semicrossed products implies orbit/order conjugacy of the underlying dynamical systems. Nest algebras and a non-free action are also considered.

The first cohomology groups of nest algebras on normed spaces

In this short note we prove that all the first cohomology groups of nest algebras acting on a normed space are trivial when the coefficients lie in any weakly closed bimodule which contains the nest algebra.

Additive derivations of nest algebras

In this paper we prove that every additive derivation of a nest algebra acting on an infinite-dimensional Hilbert space is inner. This extends the relative result for linear derivations of nest algebras.

Jacobson radicals of nest algebras in factors

Definition. Let β \beta be a nest in a separably acting type I I ∞ {\text {I}}{{\text {I}}_\infty } factor M \mathcal {M} . An element P ∈ β ∖ { 0 , I } P \in \beta \backslash \{ 0,I\} is said to be a singular point of β \beta if it satisfies either of the following conditions: (1) There is a strictly increasing sequence { Q n } ⊆ β , lim n → ∞ Q n = P \{ {Q_n}\} \subseteq \beta ,\;{\lim _{n \to \infty }}{Q_n} = P , and P − Q n P - {Q_n} is infinite for each n ∈ N n \in \mathbb {N} . Also, there is a projection Q ∈ β Q \in \beta such that Q &gt; P Q &gt; P and Q − P Q - P is finite. (2) There is a strictly decreasing sequence { Q n } ⊆ β , lim n → ∞ Q n = P \{ {Q_n}\} \subseteq \beta ,\;{\lim _{n \to \infty }}{Q_n} = P , and Q n − P {Q_n} - P is infinite for each n ∈ N n \in \mathbb {N} . Also, there is a projection Q ∈ β Q \in \beta such that Q &gt; P Q &gt; P and P − Q P - Q is finite. Main Theorem. Let β \beta be a nest in a separably acting factor M \mathcal {M} . (1) If M \mathcal {M} is of type I I ∞ {\text {I}}{{\text {I}}_\infty } , then a necessary and sufficient condition for the Jacobson radical R β {\mathcal {R}_\beta } of alg ⁡ β \operatorname {alg} \beta to be a norm-closed singly generated ideal of alg ⁡ β \operatorname {alg} \beta is that the nest β \beta is countable and it does not contain a singular point. (2) If M \mathcal {M} is of type I I 1 {\text {I}}{{\text {I}}_1} or type III {\text {III}} , then a necessary and sufficient condition for the Jacobson radical R β {\mathcal {R}_\beta } of alg ⁡ β \operatorname {alg} \beta to be a norm-closed singly generated ideal of alg ⁡ β \operatorname {alg} \beta is that the nest β \beta is countable. (3) In (1) and (2) the single generation is equivalent to countable generation.