Nest Algebras Research Articles

On the topological stable rank of non-selfadjoint operator algebras

We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu’s non-commutative disc algebras and to free semigroup algebras as well.

Generalized Jordan derivation on nest algebras

Let N be a nest on a Banach space X, and Alg N be the associated nest algebra. In this paper, we prove that, if there is a nontrivial element in N which is complemented in X, then every additive generalized Jordan derivation from Alg N into itself is an additive generalized derivation. Moreover, we give a characterization of linear generalized Jordan derivations of nest algebras on complex separable Hilbert spaces.

On the topological stable rank of non-selfadjoint operator algebras

We provide a negative solution to a question of M. Rieffel who asked if the right and left topological stable ranks of a Banach algebra must always agree. Our example is found amongst a class of nest algebras. We show that for many other nest algebras, both the left and right topological stable ranks are infinite. We extend this latter result to Popescu's non-commutative disc algebras and to free semigroup algebras as well.

All-derivable points in continuous nest algebras

Let A be an operator algebra on a Hilbert space. We say that an element G ∈ A is an all-derivable point of A for the strong operator topology if every strong operator topology continuous derivable linear mapping φ at G (i.e. φ ( S T ) = φ ( S ) T + S φ ( T ) for any S , T ∈ alg N with S T = G ) is a derivation. Let N be a continuous nest on a complex and separable Hilbert space H. We show in this paper that every orthogonal projection operator P ( M ) ( 0 ≠ M ∈ N ) is an all-derivable point of alg N for the strong operator topology.

All-derivable points of operator algebras

Let A be an operator subalgebra in B ( H ) , where H is a Hilbert space. We say that an element Z ∈ A is an all-derivable point of A for the norm-topology (strongly operator topology, etc.) if, every norm-topology (strongly operator topology, etc.) continuous derivable linear mapping φ at Z (i.e. φ ( ST ) = φ ( S ) T + S φ ( T ) for any S , T ∈ A with ST = Z ) is a derivation. In this paper, we show that every invertible operator in the nest algebra alg N is an all-derivable point of the nest algebra for the strongly operator topology. We also prove that every nonzero element of the algebra of all 2 × 2 upper triangular matrixes is an all-derivable point of the algebra.

Lie triple derivations on nest algebras

AbstractLet δ be a Lie triple derivation from a nest algebra 𝒜 into an 𝒜‐bimodule ℳ︁. We show that if ℳ︁ is a weak* closed operator algebra containing 𝒜 then there are an element S ∈ ℳ︁ and a linear functional f on 𝒜 such that δ (A) = SA – AS + f (A)I for all A ∈ 𝒜, and if ℳ︁ is the ideal of all compact operators then there is a compact operator K such that δ (A) = KA – AK for all A ∈ 𝒜. As applications, Lie derivations and Jordan derivations on nest algebras are characterized. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Jordan maps on triangular algebras

Let T be a triangular algebra and R′ be an arbitrary ring. Suppose that M : T → R ′ and M ∗ : R ′ → T are surjective maps such that M ( aM ∗ ( x ) + M ∗ ( x ) a ) = M ( a ) x + xM ( a ) , M ∗ ( M ( a ) x + xM ( a ) ) = aM ∗ ( x ) + M ∗ ( x ) a for all a ∈ T , x ∈ R ′ . In this paper, we give sufficient conditions on T such that both M and M ∗ are additive. In particular, if T is a standard subalgebra of a nest algebra, then both M and M ∗ are additive.

Derivable mappings at unit operator on nest algebras

Let A be an operator subalgebra with the unit operator I in B(H). We say that a linear mapping φ from A into itself is a derivable mapping at I if φ ( ST ) = φ ( S ) T + S φ ( T ) for any S , T ∈ A with ST = I. In this paper, we show the following main result: every strongly operator topology continuous derivable mapping at I on a nest algebra alg N is an inner derivation.

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 .

Additive Jordan derivations of reflexive algebras

Additive Jordan derivations of certain reflexive algebras are investigated. In particular, additive Jordan derivations of nest algebras on Banach spaces are shown to be additive derivations.

Generalized biderivations of nest algebras

Let N be a nest on a complex separable Hilbert space H, and τ ( N ) be the associated nest algebra. In this paper, we prove that every biderivation of τ ( N ) is an inner biderivation if and only if dim 0 + ≠ 1 or dim H - ⊥ ≠ 1 , and that every generalized biderivation of τ ( N ) is an inner generalized biderivation if dim 0 + ≠ 1 and dim H - ⊥ ≠ 1 .

Lie triple derivations of nest algebras

Let N be a nest on a complex separable Hilbert space H , and τ ( N ) be the associated nest algebra. In this paper, we prove that every Lie triple derivation from τ ( N ) into itself is of the form X → XT − TX + h( X) I, where T ∈ τ ( N ) and h is a linear mapping from τ ( N ) into C such that h([[ A, B], C]) = 0 for all A , B , C ∈ τ ( N ) .

Multiplicative Mappings of Rings

Let ℛ and \({\fancyscript S}\) be arbitrary associative rings. A mapping φ of ℛ onto \({\fancyscript S}\) is called a multiplicative isomorphism if φ is bijective and satisfies φ(xy) = φ(x)φ(y) for all x, y ∈ ℛ. In this short note, we establish a condition on ℛ, in the case where ℛ may not contain any non–zero idempotents, that assures that φ is additive, which generalizes the famous Martindale's result. As an application, we show that under a mild assumption every multiplicative isomorphism from the radical of a nest algebra onto an arbitrary ring is additive.

ADDITIVITY OF JORDAN MULTIPLICATIVE MAPS ON JORDAN OPERATOR ALGEBRAS

Let $H$ be a Hilbert space and $\mathcal{N}$ a nest in $H$. Denote by $S^a(H)$ the Jordan ring of all self-adjoint operators on $H$ and $\mathrm{Alg}\mathcal{N}$ the nest algebra associated to $\mathcal{N}$. We show that a bijective map $\Phi : S^a(H) \to S^a(H)$ satisfying (1) $\Phi(ABA) = \Phi(A) \Phi(B) \Phi(A)$ for every pair of $A,B$, or (2) $\Phi(AB + BA) = \Phi(A) \Phi(B) + \Phi(B) \Phi(A)$ for every pair of $A,B$, or (3) $\Phi(\frac{1}{2} (AB + BA)) = \frac{1}{2} (\Phi(A) \Phi(B) + \Phi(B) \Phi(A))$ for every pair of $A,B$ must be additive, that is, a Jordan ring isomorphism. We also show that if a bijective map $\Phi : \mathrm{Alg}\mathcal{N} \to \mathrm{Alg}\mathcal{N}$ satisfies the Jordan multiplicativity of the form (2) or (3), then $\Phi$ must be a Jordan isomorphism. Moreover, such Jordan multiplicative maps are characterized completely.

The Existence of Maximal w*–closed Submodules in Nest Algebra Modules

Using the technique of annihilators, we characterize the w*–closure of the finite–partition submodule ℛ(\({\fancyscript N}\),~)of a weakly closed nest algebra module \({\fancyscript U}\) completely; and by virtue of this characterization, we obtain a sufficient and necessary condition for \({\fancyscript U}\) to have a maximal w*–closed submodule.

A characteristic property of nest algebras

A characteristic property of nest algebras

Elementary maps on nest algebras

Let A 1 , A 2 be algebras and let M : A 1 → A 2 , M ∗ : A 2 → A 1 be maps. An elementary map of A 1 × A 2 is an ordered pair ( M , M ∗ ) such that { M ( A M ∗ ( B ) C ) = M ( A ) B M ( C ) , M ∗ ( B M ( A ) D ) = M ∗ ( B ) A M ∗ ( D ) for all A , C ∈ A 1 , B , D ∈ A 2 . In this paper, the general form of surjective elementary maps on standard subalgebras of nest algebras is described. In particular, such maps are automatically additive.

On the commutant and orbits of conjugation

In this note, we analyse the relationship between the commutant of a bounded linear operator A and the algebra of similarity B A that was introduced in the late 70s as a characterization of nest algebras. Necessary and sufficient conditions are also obtained for an operator to commute with real scalar generalized operators in the sense of Colojoara-Foias in Banach spaces. In the second part, we analyse the relationship between the generalized inverse, the generalized commutant and the orbits of conjugation.

Jordan isomorphisms of triangular rings

We investigate Jordan isomorphisms of triangular rings and give a sufficient condition under which they are necessarily isomorphisms or anti-isomorphisms. As corollaries we obtain generalizations of two recent results: the one concerning Jordan isomorphisms of triangular matrix algebras by Beidar, Brešar and Chebotar, and the one concerning Jordan isomorphisms of nest algebras by Lu.