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, Bresar and Chebotar, and the one concerning Jordan isomorphisms of nest algebras by Lu.

     

三角代数上初等映射的可加性

本文主要研究了三角代数上的初等映射的可加性.利用矩阵分块理论,证明了三角代数上的初等映射是可加的.     

三角代数上的Jordan零点ξ-Lie可导映射

给出了三角代数上Jordan零点ξ-Lie可导映射的结构.作为应用,得到了套代数上Jordan零点ξ-Lie可导映射的具体形式.     

Nonlinear Jordan Higher Derivations of Triangular Algebras

In this paper, we prove that any nonlinear Jordan higher derivation on triangular algebras is an additive higher derivation. As a byproduct, we obtain that any nonlinear Jordan derivation on nest algeb...     

Jordan higher derivable maps on triangular algebras by commutative zero products

In this paper, the structure of Jordan higher derivable maps on triangular algebras by commutative zero products is given. As an application, the form of Jordan higher derivable maps of nest algebras by commutative zero products is obtained.     

Jordan generalized derivations on triangular algebras

Let 𝒜 be a unital algebra and let ? be a unitary 𝒜-bimodule. We consider Jordan generalized derivations mapping from 𝒜 into ?. Our results on unitary algebras are applied to triangular algebras. In particular, we prove that any Jordan generalized derivation of a triangular algebra is a generalized derivation.     

Spectral conditions on Lie and Jordan algebras of compact operators

We investigate the properties of bounded operators which satisfy a certain spectral additivity condition, and use our results to study Lie and Jordan algebras of compact operators. We prove that these algebras have nontrivial invariant subspaces when their elements have sublinear or submultiplicative spectrum, and when they satisfy simple trace conditions. In certain cases we show that these conditions imply that the algebra is (simultaneously) triangularizable.     

三角代数上的广义Jordan导子

主要研究了三角代数上的广义Jordan导子．利用三角代数上广义Jordan导子和广义内导子的联系．证明了作用在一个含单位元的可交换环上的三角代数到其自身上的环线性广义Jordan导子是一个广义导子．     

Linear maps on operator algebras that preserve elements annihilated by a polynomial

In this paper some purely algebraic results are given concerning linear maps on algebras which preserve elements annihilated by a polynomial of degree greater than 1 and with no repeated roots and applied to linear maps on operator algebras such as standard operator algebras, von Neumann algebras and Banach algebras. Several results are obtained that characterize such linear maps in terms of homomorphisms, anti-homomorphisms, or, at least, Jordan homomorphisms.

     

Isomorphisms between sub-strongly reducible maximal triangular algebras

In this paper, we introduce the concept of sub-strongly maximal triangular algebras which form a large class of maximal triangular algebras, and prove that every algebraic isomorphism of sub-strongly maximal triangular algebras is spatially implemented, which generalizes the result by Ringrose in two respects.

     

Derived equivalences between triangular matrix algebras

In this paper, we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras and describe methods to construct tilting modules and tilting complexes inducing derived equivalences between them.     

Additivity of multiplicative maps on triangular rings

In this paper we shall give a unified technique in the discussion of the additivity of n-multiplicative automorphisms, n-multiplicative derivations, n-elementary surjective maps, and Jordan multiplicative surjective maps on triangular rings.     

Piecewise Hereditary Triangular Matrix Algebras

For any positive integer N,we clearly describe all finite-dimensional algebras A such that the upper triangular matrix algebras TN(A) are piecewise hereditary.Consequently,we describe all finite-dimensional algebras A such that their derived categories of N-complexes are triangulated equivalent to derived categories of hereditary abelian categories,and we describe the tensor algebras A (×) K[X]/(XN) for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.     

$(m,n)$-Igusa-Todorov 代数,IT-维数和三角矩阵代数

设$T$, $U$是两个Artin代数, $_U M_T$是$U$-$T$-双模.本文得到了三角矩阵代数$\Lambda=\left({\smallmatrix T&0\\ M&U \endsmallmatrix}\right)$是$(m,n)$-Igusa-Todorov的一个充要条件.我们还研究了$\Lambda$的IT维数.更具体地说,事实证明$$\max\{\ITdim T, \ITdim U\}\leqslant\ITdim \Lambda\leqslant\min\{\max\{\gldim T,\ITdim U\},\max\{\gldim U,\ITdim T\}\}.$$     

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.     

Assosymmetric algebras under Jordan product

We prove that assosymmetric algebras under the Jordan product are Lie triple algebras. A Lie triple algebra is called special if it is isomorphic to a subalgebra of the plus-algebra of some assosymmetric algebra. We establish that the Glennie identity of degree 8 is valid for special Lie triple algebras, but not for all Lie triple algebras.     

Linear Maps Preserving Idempotence on Nest Algebras

In this paper, we discuss the rank-l-preserving linear maps on nest algebras of Hilbertspace operators. We obtain several characterizations of such linear maps and apply them to show that a weakly continuous linear hijection on an atomic nest algebra is idempotent preserving if and only if it is a Jordan homomorphism, and in turn, if and only if it is an automorphism or an anti-automorphism.     

单位积决定的若当矩阵代数(英文)

本文研究了单位积决定的若当矩阵代数M=Mn(R)的条件及分类问题.利用基矩阵及巧妙对对称双线性映射{·,·}进行构造和扩充,用初等矩阵的方法,获得了一系列新的同样重要的定义,结论与证明(与参考文献[1]相比较),推广了参考文献[1]的结论,作为其应用可以进一步证明了Mn(R)上的任意可逆线性映射都是保单位积的.     

Spectral order of operators and range projections

We study the effect algebra (i.e. the positive part of the unit ball of an operator algebra) and its relation to the projection lattice from the perspective of the spectral order. A spectral orthomorphism is a map between effect algebras which preserves the spectral order and orthogonality of elements. We show that if the spectral orthomorphism preserves the multiples of the unit, then it is a restriction of a Jordan homomorphism between the corresponding algebras. This is an optimal extension of the Dye's theorem on orthomorphisms of the projection lattices to larger structures containing the projections. Moreover, results on automatic countable additivity of spectral homomorphisms are proved. Further, we study the order properties of the range projection map, assigning to each positive contraction in a JBW algebra its range projection. It is proved that this map preserves infima of positive contractions in the spectral (respectively standard order) if, and only if, the projection lattice of the algebra in question is a modular lattice.