Commutative Subspace Lattice Research Articles

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.

Multiplicative generalized Lie n-derivations on completely distributive commutative subspace lattice algebras

Multiplicative generalized Lie n-derivations on completely distributive commutative subspace lattice algebras

Linear Mappings Characterized by Action on Zero Products or Unit Products

Let $$\mathcal {A}$$ be a unital algebra and $$\mathcal {M}$$ be a unital $$\mathcal {A}$$ -bimodule. We characterize the linear mappings $$\delta $$ and $$\tau $$ from $$\mathcal {A}$$ into $$\mathcal {M}$$ , satisfying $$\delta (A)B+A\tau (B)=0$$ for every $$A,B \in \mathcal {A}$$ with $$AB=0$$ when $$\mathcal {A}$$ contains a separating ideal $$\mathcal {T}$$ of $$\mathcal {M}$$ , which is in the algebra generated by all idempotents in $$\mathcal {A}$$ . We apply the result to $$\mathcal {P}$$ -subspace lattice algebras, completely distributive commutative subspace lattice algebras, and unital standard operator algebras. Furthermore, suppose that $$\mathcal {A}$$ is a unital Banach algebra and $$\mathcal {M}$$ is a unital Banach $$\mathcal {A}$$ -bimodule, we give a complete description of linear mappings $$\delta $$ and $$\tau $$ from $$\mathcal A$$ into $$\mathcal M$$ , satisfying $$\delta (A)B+A\tau (B)=0$$ for every $$A,B\in \mathcal {A}$$ with $$AB=I$$ .

Characterizations of Jordan left derivations on some algebras

A linear mapping δ from an algebra A into a left A-module M is called a Jordan left derivation if δ(A2)=2Aδ(A) for every A∈A. We prove that if an algebra A and a left A-module M satisfy one of the following conditions—(1) A is a C∗-algebra and M is a Banach left A-module; (2) A=AlgL with ∩{L−:L∈JL}=(0) and M=B(X); and (3) A is a commutative subspace lattice algebra of a von Neumann algebra B and M=B(H)—then every Jordan left derivation from A into M is zero. δ is called left derivable at G∈A if δ(AB)=Aδ(B)+Bδ(A) for each A,B∈A with AB=G. We show that if A is a factor von Neumann algebra, G is a left separating point of A or a nonzero self-adjoint element in A, and δ is left derivable at G, then δ≡0.

Tensor Products of Subspace Lattices and Rank One Density

We show that, if $M$ is a subspace lattice with the property that the rank one subspace of its operator algebra is weak* dense, $L$ is a commutative subspace lattice and $P$ is the lattice of all projections on a separable infinite dimensional Hilbert space, then the lattice $L\otimes M\otimes P$ is reflexive. If $M$ is moreover an atomic Boolean subspace lattice while $L$ is any subspace lattice, we provide a concrete lattice theoretic description of $L\otimes M$ in terms of projection valued functions defined on the set of atoms of $M$. As a consequence, we show that the Lattice Tensor Product Formula holds for $\Alg M$ and any other reflexive operator algebra and give several further corollaries of these results.

(m,n)-Jordan derivations

A subspace lattice L on H is called commutative subspace lattice if all projections in L commute pairwise. It is denoted by CSL. If L is a CSL, then algL is called a CSL algebra. Under the assumption m + n ? 0 where m,n are fixed integers, if ? is a mapping from L into itself satisfying the condition (m + n)?(A2) = 2m?(A)A + 2nA?(A) for all A?A, we call ? an (m,n) Jordan derivation. We show that if ? is a norm continuous linear (m,n) mapping from A into it self then ? is a (m,n)-Jordan derivation.

Lie Triple Derivations of CSL Algebras

Let Open image in new window be a commutative subspace lattice generated by finite many commuting independent nests on a complex separable Hilbert space Open image in new window with Open image in new window, and Open image in new window the associated CSL algebra. It is proved that every Lie triple derivation from Open image in new window into any σ-weakly closed algebra Open image in new window containing Open image in new window is of the form X→XT−TX+h(X)I, where Open image in new window and h is a linear mapping from Open image in new window into ℂ such that h([[A,B],C])=0 for all Open image in new window.

CHARACTERIZATIONS OF HIGHER DERIVATIONS AND JORDAN HIGHER DERIVATIONS ON CSL ALGEBRAS

AbstractLet ℒ be a commutative subspace lattice and 𝒜=Alg ℒ. It is shown that every Jordan higher derivation from 𝒜 into itself is a higher derivation. We say that D=(δi)i∈ℕ is a higher derivable linear mapping at G if δn(AB)=∑ i+j=nδi(A)δj(B) for all n∈ℕ and A,B∈𝒜 with AB=G. We also prove that if D=(δi)i∈ℕ is a bounded higher derivable linear mapping at 0 from 𝒜 into itself and δn (I)=0 for all n≥1 , or D=(δi)i∈ℕ is a higher derivable linear mapping at I from 𝒜 into itself, then D=(δi)i∈ℕ is a higher derivation.

Isomorphisms and generalized derivations of some algebras

For a commutative subspace lattice L on a complex Hilbert space and a bounded bijective linear mapping h from alg L onto a unital Banach algebra B , we show that if h satisfies h ( A ) h ( B ) h ( C ) = 0 for all A , B , C in alg L with AB = BC = 0 and h ( I ) = I , then h is an isomorphism. For a J ‐ subspace lattice L on a Banach space and the unital subalgebra A of alg L generated by finite-rank operators, we show that all generalized Jordan derivations from A to any unital A ‐ bimodule are generalized derivations.

Annihilator-preserving maps, multipliers, and derivations

For a commutative subspace lattice L in a von Neumann algebra N and a bounded linear map f:N∩algL→B(H), we show that if Af(B)C=0 for all A,B,C∈N∩algL satisfying AB=BC=0, then f is a generalized derivation. For a unital C∗-algebra A, a unital Banach A-bimodule M, and a bounded linear map f:A→M, we prove that if f(A)B=0 for all A,B∈A with AB=0, then f is a left multiplier; as a consequence, every bounded local derivation from a C∗-algebra to a Banach A-bimodule is a derivation. We also show that every local derivation on a semisimple free semigroupoid algebra is a derivation and every local multiplier on a free semigroupoid algebra is a multiplier.

Lie and Jordan Ideals in Reflexive Algebras

A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper, we show, for a weakly closed linear subspace $${\mathfrak{I}}$$ of a CDCSL algebra $${\mathcal{A}}$$ , that $${\mathfrak{I}}$$ is a Lie ideal if and only if $${{A\mathfrak{I}A}}^{-1} \subseteq \mathfrak{I}$$ for all invertibles A in $${\mathcal{A}}$$ , and that $${\mathfrak{I}}$$ is a Jordan ideal if and only if it is an associative ideal.

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.

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.

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.

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