Robert Pluta, Bernard Russo
To each projection $p$ in a $C^*$-algebra $A$ we associate a family of derivations on $A$, called $p$-derivations, and relate them to the space of triple derivations on $p A (1-p)$. We then show that every derivation on a ternary ring of operators is spatial and we investigate whether every such derivation on a weakly closed ternary ring of operators is inner.
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In this paper, we investigate homomorphisms between $C^*$-ternary algebras and derivations on $C^*$-ternary algebras, associated with the following functional equation $$f(\frac{x_2-x_1}{3})+f(\frac{x_1-3x_3}{3})+f(\frac{3x_1+3x_3-x_2}{3})=f(x_1).$$ Moreover, we prove the generalized Hyers-Ulam -Rassias stability of homomorphisms in $C^*$-ternary algebras and of derivations on $C^*$-ternary algebras.
We prove that an operator space is completely isometric to a ternary ring of operators if and only if the open unit balls of all of its matrix spaces are bounded symmetric domains. From this we obtain an operator space characterization of C*-algebras.
Ideal Spaces of the Haagerup Tensor Product of Ternary Rings of Operators
本文刻画了算子三元环V与C*-代数B的Haagerup张量积V⊗^h B的本原、因子及Glimm理想，推进了该张量积理想空间的研究。
Suppose that $X$ is a (real or complex) Banach space, $dimX \geq 2$, and $\mathcal{N}$ is a nest on $X$, with each $N$ in $\mathcal{N}$ is complemented in $X$ whenever $N_{-}=N$. A ternary derivation of $Alg\mathcal{N}$ is a triple of linear maps $(γ, δ, τ)$ of $Alg\mathcal{N}$ such that $γ(AB)=δ(A)B +Aτ(B)$ for all $A,B \in Alg\mathcal{N}$. We show that for linear maps $δ$, $τ$ on $Alg\mathcal{N}$ there exists a unique linear map $γ$ from $Alg\mathcal{N}$ into $Alg\mathcal{N}$ defined by $γ(A)=RA+AT$ for some $R$, $T$ in $Alg\mathcal{N}$ such that $(γ, δ, τ)$ is a ternary derivation of $Alg\m…
We are interested in properties, especially injectivity (in the sense of category theory), of the ternary rings of operators generated by certain subsets of an inverse semigroup via the regular representation. We determine all subsets of the extended bicyclic semigroup which are closed under the triple product $xy^*z$ (called semiheaps) and show that the weakly closed ternary rings of operators generated by them (W*-TROs) are injective operator spaces.
It is shown that if T is a ternary ring of operators (TRO), X is a nondegenerate sub-TRO of T and there exists a contractive idempotent surjective map P:T-->X, then P has a unique, explicitly described extension to a conditional expectation between the associated linking algebras. A version of the result for W*-TROs is also presented and some applications mentioned.
Actions of locally compact groups and quantum groups on W*-ternary rings of operators are discussed and related crossed products introduced. The results generalise those for von Neumann algebraic actions with proofs based mostly on passing to the linking von Neumann algebra. They are motivated by the study of fixed point spaces for convolution operators generated by contractive, non-necessarily positive measures, both in the classical and in the quantum context.
We show that there is a functor from the category of positive admissible ternary rings to the category of $*$-algebras, which induces an isomorphism of partially ordered sets between the families of $C^*$-norms on the ternary ring and its corresponding $*$-algebra. We apply this functor to obtain Morita-Rieffel equivalence results between cross sectional $C^*$-algebras of Fell bundles, and to extend the theory of tensor products of $C^*$-algebras to the larger category of full Hilbert $C^*$-modules. We prove that, like in the case of $C^*$-algebras, there exist maximal and minimal tensor produ…
The ideal structure of the minimal Tensor Product of Ternary Rings of Operators
研究三元算子环与C*-代数的算子空间内射张量积的理想空间结构，通过两个映射刻画其原理想与素理想，证明在核-核拓扑下连续并诱导同胚，并给出最小素理想空间的乘积分解。
A well known result of Haagerup from 1983 states that every C*-algebra A is weakly amenable, that is, every (associative) derivation from A into its dual is inner. A Banach algebra B is said to be ternary weakly amenable if every continuous Jordan triple derivation from B into its dual is inner. We show that commutative C*-algebras are ternary weakly amenable, but that B(H) and K(H) are not, unless H is finite dimensional. More generally, we inaugurate the study of weak amenability for Jordan Banach triples, focussing on commutative JB*-triples and some Cartan factors.
In the present paper derivations and *-automorphisms of algebras of unbounded operators over the ring of measurable functions are investigated and it is shown that all L^0-linear derivations and L^{0}-linear *-automorphisms are inner. Moreover, it is proved that each L^0-linear automorphism of the algebra of all linear operators on a bo-dense submodule of a Kaplansky-Hilbert module over the ring of measurable functions is spatial.
Anti-C*-algebras
本文定义反 C*-代数，揭示其与 C*-三元环赋范标准嵌入的直和结构，证明二者半单，并给出同构于 TRO 的 C*-三元环的两个新刻画，回答 Zettl 于 1983 年提出的问题。
In the present paper we prove that every 2-local inner derivation on the matrix ring over a commutative ring is an inner derivation and every derivation on an associative ring has an extension to a derivation on the matrix ring over this associative ring. We also develop a Jordan analog of the above method and prove that every 2-local inner derivation on the Jordan matrix ring over a commutative ring is a derivation.
Operator Categories I. T*-categories
引入T*-范畴作为C*-范畴的三元推广，构造其连结C*-范畴，将Zettl的Gelfand-Naimark表示定理推广至T*-范畴，并探讨其与C*-范畴的对偶。
In the present paper it is proved that every inner 2-local derivation on the matrix ring $M_n(\Re)$ of $n\times n$ matrices over a commutative associative ring $\Re$ is an inner derivation. Also, it is proved that, every derivation on an associative ring $\Re$ has an extension to a derivation on the matrix ring $M_n(\Re)$ of $n\times n$ matrices over $\Re$.
In this paper we generalize the result valid for associative rings due \cite[Martindale III]{Mart} and \cite[Bre$\check{s}$ar]{bresar} to alternative rings. Let $\mathfrak{R}$ be an unital alternative ring, and $\mathfrak{D}: \mathfrak{R} \rightarrow \mathfrak{R}$ is a Lie multiplicative derivation. Then $\mathfrak{D}$ is the form $δ+ τ$ where $δ$ is an additive derivation of $\mathfrak{R}$ and $τ$ is a map from $\mathfrak{R}$ into its center $\mathcal{Z}(\mathfrak{R})$, which maps commutators into the zero.
We prove that if M is a von Neumann algebra whose abelian summand is discrete, then every local derivation on the algebra of all measurable operators affilated with M is a derivation. This answers a question of Richard Kadison.
2-Local and local derivations on Jordan matrix rings over commutative involutive rings
研究证明了在交换反环上的自共轭矩阵的乔丹环上，所有2-局部内导数都是导数，并应用于无限维自共轭矩阵值映射的乔丹代数，证明了所有2-局部空间导数都是空间导数。
In the present paper we prove that every 2-local inner derivation on the Lie ring of skew-symmetric matrices over a commutative ring is an inner derivation. We also apply our technique to various Lie algebras of infinite dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation.
In the present paper we prove that every additive (not necessarily homogenous) local inner derivation on the algebra of matrices over an arbitrary field is an inner derivation, and every local inner derivation on the ring of matrices over a finite ring generated by the identity element or the ring of integers is an inner derivation. We also prove that every additive local inner derivation on the Jordan algebra of symmetric matrices over an arbitrary field is a derivation, and every local inner derivation on the Jordan ring of symmetric matrices over a finite ring generated by the identity elem…
Derivations of group rings for finite and FC groups
本文建立了群环导子的分解定理，利用拓扑技术研究并描述了有限群与FC群群环的所有导子，还给出了有限群情形下显式计算导子的算法。
A map $ϕ$ on an associative ring is called a multiplicative Lie derivation if $ϕ([x,y])=[ϕ(x),y]+[x,ϕ(y)]$ holds for any elements $x,y$, where $[x,y]=xy-yx$ is the Lie product. In the paper, we discuss the multiplicative Lie derivations on the triangular 3-matrix rings $\mathcal T={\mathcal T}_3(\mathcal R_i; \mathcal M_{ij})$. Under the standard assumption $Q_i\mathcal Z(\mathcal T)Q_i=\mathcal Z(Q_i\mathcal T Q_i)$, $i=1,2,3$, we show that every multiplicative Lie derivation $φ:\mathcal T\to\mathcal T$ has the standard form $φ=δ+γ$ with $δ$ a derivation and $γ$ a center valued map vanishing …
A general form of contractive idempotent functionals on coamenable locally compact quantum groups is obtained, generalising the result of Greenleaf on contractive measures on locally compact groups. The image of a convolution operator associated to a contractive idempotent is shown to be a ternary ring of operators. As a consequence a one-to-one correspondence between contractive idempotents and a certain class of ternary rings of operators is established.
A combinatorial view on derivations in bimodules
本文利用与广群特征空间相关的方法研究群环上双模中的导子，证明了描述导子空间的定理，并探讨了包括$(σ, τ)$-导子在内的具体例子。
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