CSL Algebras Research Articles

Norms of vector functionals

We examine the question of when, and how, the norm of a vector functional on an operator algebra can be controlled by the invariant subspace lattice of the algebra. We introduce a related operator algebraic property, and show that it is satisfied by all von Neumann algebras and by all CSL algebras. We exhibit examples of operator algebras that do not satisfy the property or any scaled version of it.

Crossed products of operator algebras: Applications of Takai duality

Let (G,Σ) be a (partially) ordered abelian group with Haar measure μ, let (A,G,α) be a dynamical system and let A⋊αΣ be the associated semicrossed product. Using Takai duality we establish a stable isomorphismA⋊αΣ∼s(A⊗K(G,Σ,μ))⋊α⊗AdρG, where K(G,Σ,μ) denotes the compact operators in the CSL algebra AlgL(G,Σ,μ) and ρ denotes the right regular representation of G. We also show that there exists a complete lattice isomorphism between the αˆ-invariant ideals of A⋊αΣ and the (α⊗Adρ)-invariant ideals of A⊗K(G,Σ,μ).Using Takai duality we also continue our study of the Radical for the crossed product of an operator algebra and we solve open problems stemming from the earlier work of the authors. Among others we show that the crossed product of a radical operator algebra by a compact abelian group is a radical operator algebra. We also show that the permanence of semisimplicity fails for crossed products by R. A final section of the paper is devoted to the study of radically tight dynamical systems, i.e., dynamical systems (A,G,α) for which the identity Rad(A⋊αG)=(RadA)⋊αG persists. A broad class of such dynamical systems is identified.

(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.

Normalisers of operator algebras and tensor product formulas

We establish a tensor product formula for bimodules over maximal abelian self-adjoint algebras and their supports. We use this formula to show that if \mathcal{A} is the tensor product of finitely many continuous nest algebras, \mathcal{B} is a CSL algebra and \mathcal{A} and \mathcal{B} have the same normaliser semigroup then either \mathcal{A} =\mathcal{B} or \mathcal{ A}^* = \mathcal{B} . We show that the result does not hold without the assumption that the nests be continuous, answering in the negative a question previously raised in the literature.

Characterization of derivations on reflexive algebras

Let Algℒ be a reflexive algebra on a Hilbert space H. We say that a linear map δ: Algℒ → Algℒ is derivable at Ω ∈ Algℒ if δ(A)B + Aδ(B) = δ(Ω) for every A, B ∈ Algℒ with AB = Ω. In this article, we give a necessary and sufficient condition for a map δ on Algℒ to be derivable at Ω. In particular, we show that every linear map δ derivable at Ω ≠ 0 from an irreducible CDC algebra (in particular, a nest algebra) into itself is a derivation. Moreover, if Algℒ is a CSL algebra, and if for some nontrivial projection P ∈ ℒ, PΩP and (I − P)Ω(I − P) are left or right separating points in PAlgℒP and (I − P)Algℒ(I − P) respectively, then a linear map δ on Algℒ is derivable at Ω if and only if δ is a 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.

Characterization of higher derivations on CSL algebras

Let AlgL be a CSL algebra. We say that a family of linear maps δ={δn,δn:AlgL→AlgL,n∈N} is higher derivable at Ω∈AlgL if ∑i+j=nδi(A)δj(B)=δn(Ω) for all A,B∈AlgL with AB=Ω. In this paper, a necessary and sufficient condition for a family of linear maps δ={δn,n∈N} on AlgL to be higher derivable at Ω∈AlgL is given. Moreover, we show that if there is a faithful projection P in L such that PΩP and (I−P)Ω(I−P) are a left or right separating point in PAlgLP and (I−P)AlgL(I−P) respectively, then a family of linear maps δ={δn,n∈N} on AlgL is higher derivable at Ω if and only if it is a higher derivation. In particular, if AlgL is an irreducible CDCSL algebra or a nest algebra, then a family of linear maps δ={δn,n∈N} on AlgL is higher derivable at Ω≠0 if and only if it is a higher derivation.

Generalized derivations associated with Hochschild 2-cocycles on some algebras

We investigate a new type of generalized derivations associated with Hochschild 2-cocycles which was introduced by A. Nakajima. We show that every generalized Jordan derivation of this type from CSL algebras or von Neumann algebras into themselves is a generalized derivation under some reasonable conditions. We also study generalized derivable mappings at zero point associated with Hochschild 2-cocycles on CSL algebras.

Stable isomorphism of dual operator spaces

We prove that two dual operator spaces X and Y are stably isomorphic if and only if there exist completely isometric normal representations ϕ and ψ of X and Y, respectively, and ternary rings of operators M 1 , M 2 such that ϕ ( X ) = [ M 2 ∗ ψ ( Y ) M 1 ] − w ∗ and ψ ( Y ) = [ M 2 ϕ ( X ) M 1 ∗ ] − w ∗ . We prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic. We apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic. Consequently, we obtain that certain complex domains are biholomorphically equivalent if and only if their algebras of bounded analytic functions are Morita equivalent in our sense. Finally, we provide examples motivated by the theory of CSL algebras.

The Jordan structure of CSL algebras

We show that every Jordan isomorphism between CSL algebras is the sum of an isomorphism and an anti-isomorphism. Also we show that each Jordan derivation of a CSL algebra is a derivation.

Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if A 1 and A 2 are operator algebras, then any bounded epimorphism of A 1 onto A 2 is completely bounded provided that A 2 contains a norming C*-subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a C*-algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a C*-algebra is similar to a *-representation precisely when the image operator algebra A-norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras A i of C*-diagonals (C i ,D i ) (i = 1, 2) satisfying D i C A i C C, extends uniquely to a *-isomorphism of the C*-algebras generated by A 1 and A 2 ; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.

Completely bounded cohomology of non-selfadjoint operator algebras

The authors prove that all n-th completely bounded cohomology groups of a nest algebra T( N) acting on a separable Hilbert space are trivial when the coefficients lie in any ultraweakly closed T( N)-bimodule containing the nest algebra. They also prove that H c b n ( A , M ) ≅ H c b n ( A , A ) for all n ≥ 1 and a CSL algebra A with an ultraweakly closed A-bimodule M containing A.

Characterizations of isomorphisms and derivations of some algebras

Let ϕ be a zero-product preserving bijective bounded linear map from a unital algebra A onto a unital algebra B such that ϕ ( 1 ) = k . We show that if A is a CSL algebra on a Hilbert space or a J -lattice algebra on a Banach space then there exists an isomorphism ψ from A onto B such that ϕ = k ψ . For a nest algebra A in a factor von Neumann algebra, we characterize the linear maps on A such that δ ( x ) y + x δ ( y ) = 0 for all x , y ∈ A with x y = 0 .

Lie derivations of certain CSL algebras

It is shown that each Lie derivation on a reflexive algebra, whose lattice is completely distributive and commutative, can be uniquely decomposed into the sum of a derivation and a linear mapping with image in the center of the algebra.

Lie isomorphisms of reflexive algebras

A Lie isomorphism ϕ between algebras is called trivial if ϕ = ψ + τ , where ψ is an (algebraic) isomorphism or a negative of an (algebraic) anti-isomorphism, and τ is a linear map with image in the center vanishing on each commutator. In this paper, we investigate the conditions for the triviality of Lie isomorphisms from reflexive algebras with completely distributive and commutative lattices (CDCSL). In particular, we prove that a Lie isomorphism between irreducible CDCSL algebras is trivial if and only if it preserves I-idempotent operators (the sum of an idempotent and a scalar multiple of the identity) in both directions. We also prove the triviality of each Lie isomorphism from a CDCSL algebra onto a CSL algebra which has a comparable invariant projection with rank and corank not one. Some examples of Lie isomorphisms are presented to show the sharpness of the conditions.

Local derivations on certain CSL algebras

In this paper, it is shown that every norm continuous linear local derivation from an arbitrary CSL algebra whose lattice is generated by finitely many independent nests into any ultraweakly closed subalgebra which contains the algebra is an inner derivation, and that every norm continuous linear local derivation from an arbitrary CSL algebra whose lattice is completely distributive into any ultraweakly closed subalgebra which contains the algebra is a derivation.

Derivations of CSL algebras

We introduce the notion of quiltings for derivations acting on a CSL algebra A. A quilting is a family of pairs (X α , E α ), where E α is an invariant subspace, A α implements the derivation on the compression of the algebra to E α , and the norms of the X 's are bounded. We show that many derivations allow quiltings. As a byproduct, we can show that some derivations are spatially implemented, even when they act on algebras whose lattices are not completely distributive. If the identity in the algebra can be approximated by invariant projections E α so that (E α )- ¬= I, then every derivation from Alg £ is spatially implemented.

Generalized derivable mappings at zero point on some reflexive operator algebras

Let A be a subalgebra with the unit operator I in B( H), we say that a linear mapping ϕ from A into B( H) is a generalized derivable mapping at zero point if ϕ( ST) = ϕ( S) T + Sϕ( T) − Sϕ( I) T for any S, T ∈ A with ST = 0. In this paper, we show the following main result: every norm-continuous generalized derivable mapping at zero point on finite CSL algebras is a generalized derivation.

Normalisers of CSL algebras

A coordinate description of the normalisers of CSL algebras is provided. It is shown that the linear span of the normalisers of a continuous nest algebra A, which lie in A, is ultraweakly dense in A. It is proved that, within the classes of continuous nest algebras and totally atomic CSL algebras, the normaliser semigroup determines the algebra up to adjoint. A complete description of the relation between two nest algebras with the same normalisers is given.

Isomorphisms for CSL Algebras

We study isomorphisms mapping one CSL algebra to another. An isomorphism is spatial if it is implemented by a similarity transformation given by a bounded invertible operator. An isomorphism is quasi-spatial if the implementing operator is densely defined and one-to-one, but not necessarily bounded. In an earlier paper, Frank Gilfeather and the current author showed that if £ 1 and £ 2 are commutative and £ 1 is completely distributive, and if there is a sequence E n of projections in Alg£ 1 such that E n → I and (E n ) - ¬= I, then every isomorphism from Alg £ 1 onto Alg £ 2 is quasi-spatial. In this paper, we prove a theorem with the same conclusion, but without the hypothesis of complete distributivity. Along the way, we introduce the notion of a quilting. An isomorphism which is not implemented spatially or quasi-spatially may still have a quilting, which is a way of locally implementing the isomorphism on subspaces in the lattice.