Commutative Subspace Research Articles

Local derivations and local automorphisms

In this paper, we show that if L is a completely distributive commutative subspace lattice or a J -subspace lattice, then the space of all bounded derivations of alg L is reflexive. We also study when local automorphisms on some algebras are automorphisms.

UNITARY INTERPOLATION PROBLEMS IN CSL-ALGEBRA ALGL

Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx=y. An interpolating operator for n-vectors satisfies the equation Ax<TEX>$_{i}$</TEX>=y<TEX>$_{i}$</TEX>. for i=1,2, …, n. In this article, we investigate unitary interpolation problems in CSL-Algebra AlgL : Let L be a commutative subspace lattice on a Hilbert space H. Let x and y be vectors in H. When does there exist a unitary operator A in AlgL such that Ax=y?

Spectral Synthesis and Masa‐Bimodules

The interaction between harmonic analysis and operator theory has been fruitful. In [1] Arveson related results in the theory of operator algebras to spectral synthesis. He defined synthesis for subspace lattices and proved that certain classes of lattices are synthetic. The main result in this paper is a generalization of a result of Arveson for the case of subspace maps. In order to describe in more detail the content of the present work, we need to introduce some definitions and facts from [2] and [9].Let H1 and H2 be separable complex Hilbert spaces, and Pi be the lattice of all (orthogonal) projections on Hi, i = 1, 2. Following Erdos [1], we let M(P1, P2) denote the set of all maps φ : P1 → P2 which are 0‐preserving and v‐continuous (that is, they preserve arbitrary suprema). We will call such maps subspace maps. It was shown in [2] that each φ ∈ M(P1, P2) uniquely defines semi‐lattices S1φ ⊆ P1 and S2φ = φ(P1) ⊆ P2 such that φ is a bijection between S1φ and S2φ and is uniquely determined by its restriction to S1φ. Moreover, S1φ is meet‐complete and contains the identity projection while S2φ is join‐complete and contains the zero projection. If S1φ and S2φ are commutative, we say that φ is a commutative subspace map.

Decomposability of Finite-Rank Operators in Commutative Subspace Lattice Algebras

In this paper, we explore conditions for a finite-rank operator in a commutative subspace lattice (CSL) algebra to be decomposable (that is, it can be written as the sum of rank one operators in that algebra). We introduce correlation coefficients for rank two operators, the property (F), and correlation matrices for finite-rank operators, based on which we prove that a rank two operator is decomposable if and only if it has only finitely many correlation coefficients, and if a finite-rank operator has the property (F) (has only finitely many correlation matrices) then it is decomposable.

Single elements of finite CSL algebras

An element s s of an (abstract) algebra A {\mathcal {A}} is a single element of A {\mathcal {A}} if a s b = 0 asb=0 and a , b ∈ A a,b\in {\mathcal {A}} imply that a s = 0 as=0 or s b = 0 sb=0 . Let X X be a real or complex reflexive Banach space, and let B {\mathcal {B}} be a finite atomic Boolean subspace lattice on X X , with the property that the vector sum K + L K+L is closed, for every K , L ∈ B K,L\in {\mathcal {B}} . For any subspace lattice D ⊆ B {\mathcal {D}}\subseteq {\mathcal {B}} the single elements of Alg D {\mathcal {D}} are characterised in terms of a coordinatisation of D {\mathcal {D}} involving B {\mathcal {B}} . (On separable complex Hilbert space the finite distributive subspace lattices D {\mathcal {D}} which arise in this way are precisely those which are similar to finite commutative subspace lattices. Every distributive subspace lattice on complex, finite-dimensional Hilbert space is of this type.) The result uses a characterisation of the single elements of matrix incidence algebras, recently obtained by the authors.

Rank One Subspaces of Bimodules over Maximal Abelian Selfadjoint Algebras

Spaces of operators that are left and right modules over maximal abelian selfadjoint algebras (masa bimodules for short) are natural generalizations of algebras with commutative subspace lattices. This paper is concerned with density properties of finite rank operators and of various classes of compact operators in such modules. It is shown that every finite rank operator of a norm closed masa bimodule M is in the trace norm closure of the rank one subspace of M. An important consequence is that the rank one subspace of a strongly reflexive masa bimodule (that is, one which is the reflexive hull of its rank one operators) is dense in the module in the weak operator topology. However, in contrast to the situation for algebras, it is shown that such density need not hold in the ultraweak topology. A new method of representing masa bimodules is introduced. This uses a novel concept of anω-topology. With the appropriate notion ofω-support, a correspondence is established between reflexive masa bimodules and theirω-supports. It is shown that, if a C2-closed masa bimodule contains a trace class operator then it must contain rank one operators; indeed, every such operator is in the C2-norm closure of the rank one subspace of the module. Consequently the weak closure of any masa bimodule of trace class operators is strongly reflexive. However, the trace norm closure of the rank one subspace need not contain all trace class operators of the module. Also, it is shown that there exists a CSL algebra which contains no trace class operators yet contains an operator belonging to Cpfor allp>1. From this it follows that a transitive bimodule spanned by the rank one operators it contains need not be dense in Cpfor 1⩽p<∞. As an application, it is shown that there exists a commutative subspace lattice L such that L is non-synthetic but every weakly closed algebra which contains a masa and has invariant lattice L coincides with Alg L.

On invertibility in non-selfadjoint operator algebras

Let ? be a complete commutative subspace lattice on a Hilbert space. When ? is purely atomic, we give a necessary and sufficient condition for a(T) = o,(T) for every T in algI, where oc(T) and a(T) denote the spectrum of T in algI and B(H) respectively. In addition, we discuss the properties of the spectra and the invertibility conditions for operators in algI.

Factorization along commutative subspace lattices

A positive invertible operatorT is said to be factorable along a commutative subspace latticeL if there is an invertible operatorA inAlgL whose inverse is also inAlgL and such thatT=A*A. We investigate a number of conditions that are equivalent to factorability of a given operator along a latticeL. As a byproduct, we derive a condition that guarantees that the latticeTL, defined as {range(TE) ∶E ∈L} is commutative. Applications are suggested to the particular case of factoringL∞ functions via analytic Toeplitz operators on the polydisc.

Completely distributive CSL algebras with no complements in $\mathcal {C}_p$

Anoussis and Katsoulis have obtained a criterion for the space $\operatorname {Alg} \mathcal L\cap \mathcal C_p$ to have a closed complement in $\mathcal C_p$, where $\mathcal L$ is a completely distributive commutative subspace lattice. They show that, for a given $\mathcal L$, the set of $p$ for which this complement exists forms an interval whose endpoints are harmonic conjugates. Also, they establish the existence of a lattice $\mathcal L$ for which $\operatorname {Alg} \mathcal L\cap \mathcal C_p$ has no complement for any $p\not =2$. However, they give no specific example. In this note an elementary demonstration of a simple example of this phenomenon is given. From this it follows that for a wide range of lattices $\mathcal L$, $\operatorname {Alg} \mathcal L\cap \mathcal C_p$ fails to have a complement for any $p\not =2$.

Approximate unitary equivalence of completely distributive commutative subspace lattices

In this note, we classify completely distributive CSLs up to approximate unitary equivalence. Our proof uses a new characterization of complete distributivity and leads to a generalization of a result of Arveson on ordered group lattices. As consequences of our results, we obtain a similarity theorem for hyperreflexive, completely distributive CSLs and some new perturbation results.

A Problem in Linear Matrix Approximation

AbstractLet A be a normal operator in ℬ︁(H), H a complex Hilbert space, and let ℛ A = ≷ {AX ‐ XA:X ∈ ℬ︁(H)} be the commutator subspace of ℬ︁(H) associated with A. If B in ℬ︁(H) commutes with A, then B is orthogonal to ℛA with respect to the spectral norm; i.e., the null operator is an element of best approximation of B in ℛ A. This was proved by J. Anderson in 1973 and extended by P. J. Maher with respect to the Schatten p‐norm recently.We take a look at their result from a more approximation theoretical point of view in the finite dimensional setting; in particular, we characterize all elements of best approximation of B in RA and prove that the metric projection of H onto ℛA is continuous.

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].

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

Trace-Class Operators in CSL Algebras

AbstractIn this note we show that if 𝓛 is a commutative subspace lattice, then every trace-class operator in Alg 𝓛 lies in the norm-closure of the span of rank-one operators in Alg 𝓛. We also give an elementary proof of a recent result of Davidson and Pitts that if 𝓛 is a CSL generated by completely distributive lattice and finitely many commuting chains, then 𝓛 is compact in the strong operator topology if and only if 𝓛 is completely distributive.

Reflexivity of Commutative Subspace Lattices

A short proof is given of Arveson’s reflexivity theorem for strongly closed commutative subspace lattices.

Complemented Subspaces of Cp Spaces and CSL Algebras

In this article we consider the question of whether or not a subspace of the form Cp ∩ Alg L, where L is a commutative subspace lattice (CSL), has a complement in Cp, 1 <p < ∞. While the answer is affirmative in the case where L is a nest [8] we show that this need not be true even for completely distributive CSLs. Consequently, we consider CSLs generated by a completely distributive CSL and finitely many commuting nests and we prove that Cp ∩ Alg L has a complement in Cp if and only if U2(Cp ∩ Cq) ⊆ Cp ∩ Cq, where 1/p+1/q = 1 and U2 denotes the orthogonal projection from C2 onto C2 ∩ Alg L Moreover, we observe that this result fails for CSLs which do not satisfy the above requirement. Finally, we show that in certain cases the existence of a complement for Cp ∩ Alg L is related to the behaviour of the net {UF}F. Thus we explore the theory of integration with respect to an arbitrary CSL and we obtain results which contrast the corresponding theory for nests.

Reflexivity of commutative subspace lattices

A short proof is given of Arveson’s reflexivity theorem for strongly closed commutative subspace lattices.

Strictly cyclic operator algebras

We prove several results about the lattice of invariant subspaces of general strictly cyclic and strongly strictly cyclic operator algebras. A reflexive operator algebra A A with a commutative subspace lattice is strictly cyclic iff Lat ⁡ ( A ) ⊥ \operatorname {Lat}{(A)^ \bot } contains a finite number of atoms and each nonzero element of Lat ⁡ ( A ) ⊥ \operatorname {Lat}{(A)^ \bot } contains an atom. This leads to a characterization of the n n -strictly cyclic reflexive algebras with a commutative subspace lattice as well as an extensive generalization of D. A. Herrero’s result that there are no triangular strictly cyclic operators. A reflexive operator algebra A A with a commutative subspace lattice is strongly strictly cyclic iff Lat ⁡ ( A ) \operatorname {Lat}(A) satisfies A.C.C. The distributive lattices which are attainable by strongly strictly cyclic reflexive algebras are the complete sublattices of { 0 , 1 ] × { 0 , 1 } × ⋯ \{ 0,1] \times \{ 0,1\} \times \cdots which satisfy A.C.C. We also show that if Alg ⁡ ( L ) \operatorname {Alg}(\mathcal {L}) is strictly cyclic and L ⊆ \mathcal {L} \subseteq atomic m.a.s.a. then Alg ⁡ ( L ) \operatorname {Alg}(\mathcal {L}) contains a strictly cyclic operator.

Compactness and Complete Distributivity for Commutative Subspace Lattices

Compactness and Complete Distributivity for Commutative Subspace Lattices

Arveson Nests and Operator Factorization Along Commutative Subspace Lattices

Arveson Nests and Operator Factorization Along Commutative Subspace Lattices