LIE-TYPE DERIVATIONS OF NEST ALGEBRAS ON BANACH SPACES AND RELATED TOPICS

Let $\mathcal {X}$ be a Banach space over the complex field $\mathbb {C}$ and $\mathcal {B(X)}$ be the algebra of all bounded linear operators on $\mathcal {X}$ . Let $\mathcal {N}$ be a nontrivial nest on $\mathcal {X}$ , $\text {Alg}\mathcal {N}$ be the nest algebra associated with $\mathcal {N}$ , and $L\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ be a linear mapping. Suppose that $p_n(x_1,x_2,\ldots ,x_n)$ is an $(n-1)\,$ th commutator defined by n indeterminates $x_1, x_2, \ldots , x_n$ . It is shown that L satisfies the rule $$ \begin{align*}L(p_n(A_1, A_2, \ldots, A_n))=\sum_{k=1}^{n}p_n(A_1, \ldots, A_{k-1}, L(A_k), A_{k+1}, \ldots, A_n) \end{align*} $$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ if and only if there exist a linear derivation $D\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ and a linear mapping $H\colon \text {Alg}\mathcal {N}\longrightarrow \mathbb {C}I$ vanishing on each $(n-1)\,$ th commutator $p_n(A_1,A_2,\ldots , A_n)$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ such that $L(A)=D(A)+H(A)$ for all $A\in \text {Alg}\mathcal {N}$ . We also propose some related topics for future research.

A Class of Nonlinear Nonglobal Semi-Jordan Triple Derivable Mappings on Triangular Algebras

In this paper, we proved that each nonlinear nonglobal semi-Jordan triple derivable mapping on a 2-torsion free triangular algebra is an additive derivation. As its application, we get the similar conclusion on a nest algebra or a 2-torsion free block upper triangular matrix algebra, respectively.

Jordan {g,h}-derivations on triangular algebras

In this article, we give a sufficient and necessary condition for every Jordan {g,h}-derivation to be a {g,h}-derivation on triangular algebras. As an application, we prove that every Jordan {g,h}-derivation on \tau ({\mathscr{N}}) is a {g,h}-derivation if and only if \dim {0}_{+}\ne 1 or \dim {H}_{-}^{\perp }\ne 1 , where {\mathscr{N}} is a non-trivial nest on a complex separable Hilbert space H and \tau ({\mathscr{N}}) is the associated nest algebra.

On the primitive ideals of nest algebras

We show that Ringrose's diagonal ideals are primitive ideals in a nest algebra (subject to the continuum hypothesis). This answers an old question of Lance and provides for the first time concrete descriptions of enough primitive ideals to obtain the Jacobson radical as their intersection. Separately, we provide a standard form for all left ideals of a nest algebra, which leads to insights into the maximal left ideals. In the case of atomic nest algebras, we show how primitive ideals can be categorized by their behaviour on the diagonal and provide concrete examples of all types.

Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

Let Alg 𝒩 be a nest algebra associated with the nest 𝒩 on a (real or complex) Banach space 𝕏. Suppose that there exists a non-trivial idempotent P ∈ Alg 𝒩 with range P (𝕏) ∈ 𝒩, and δ : Alg 𝒩 → Alg 𝒩 is a continuous linear mapping (generalized) left derivable at P, i.e. δ (ab) = aδ (b) + bδ (a) (δ (ab) = aδ(b) + bδ(a) − baδ(I)) for any a, b ∈ Alg 𝒩 with ab = P, where I is the identity element of Alg 𝒩. We show that is a (generalized) Jordan left derivation. Moreover, in a strongly operator topology we characterize continuous linear maps on some nest algebras Alg 𝒩 with the property that δ (P ) = 2Pδ (P ) or δ (P ) = 2P δ (P ) − Pδ (I) for every idempotent P in Alg 𝒩.

Nonlinear Isometries on Schatten-pClass in Atomic Nest Algebras

LetHbe a complex Hilbert space; denote by Alg 𝒩and𝒞p(H)the atomic nest algebra associated with the atomic nest𝒩onHand the space of Schatten-pclass operators on,Hrespectively. Let𝒞p(H)∩Alg 𝒩be the space of Schatten-pclass operators in Alg 𝒩. When1≤p<+∞andp≠2, we give a complete characterization of nonlinear surjective isometries on𝒞p(H)∩Alg 𝒩. Ifp=2, we also prove that a nonlinear surjective isometry on𝒞2(H)∩Alg 𝒩is the translation of an orthogonality preserving map.

Morita Equivalence of Nest Algebras

Let $\mathscr{N}_1$ (resp. $\mathscr{N}_2$) be a nest, $A$ (resp. $B$) be the corresponding nest algebra, $A_0$ (resp. $B_0$) be the subalgebra of compact operators. We prove that the nests $\mathscr{N}_1, \mathscr{N}_2$ are isomorphic if and only if $A$ and $B$ are weakly-$*$ Morita equivalent if and only if $A_0$ and $ B_0$ are strongly Morita equivalent. We characterize the nest isomorphisms which implement stable isomorphism between the corresponding nest algebras.