Abstract. In this paper, we show that under certain conditions every Lie higher derivation and Lie triple derivation on a triangular algebra are proper, respectively. The main results are then applied to (block) upper triangular matrix algebras and nest algebras.

In this paper, we characterize Jordan derivable mappings in terms of Peirce decomposition and determine Jordan all-derivable points for some general bimodules. Then we generalize the results to the case of Jordan higher derivable mappings. An immediate application of our main results shows that for a nest N on a Banach X with the associated nest algebra algN , if there exists a non-trivial element in N which is complemented in X, then every C ∈ algN is a Jordan all-derivable point of L(algN , B(X)) and a Jordan higher all-derivable point of L(algN ). the relations among derivations, Jordan derivations as well as inner derivations (see for example [3,4,10,14,23,28,32], and the references therein).In general there are two directions in the study of the local actions of derivations of operator algebras. One is the well known local derivation problem (see for example [7,9,15,35,40]).The other is to study conditions under which derivations of operator algebras can be completely determined by the action on some subsets of operators (see for example [3,5,17,30,34,39]).obvious that a linear mapping is a Jordan derivation if and only if it is Jordan derivable at all points. It is natural and interesting to ask the question whether or not a linear mapping is a Jordan derivation if it is Jordan derivable only at one given point. If such a point exists, we call this point a Jordan all-derivable point. To be more precise, an element C ∈ A is called a Jordan all-derivable point of L(A, M) if every Jordan derivable mapping at C is a Jordan derivation. It is quite surprising that there do exist Jordan all-derivable points for some algebras. An and Hou [2] show that under some mild conditions on unital prime ring or triangular ring A, I is a Jordan all-derivable point of L(A). Jiao and Hou [13] study Jordan derivable mappings at zero point on nest algebras. Zhao and Zhu [33] prove that 0 and I are Jordan all-derivable points of the triangular algebra. In [16], the authors study some derivable mappings in the generalized matrix algebra A, and show that 0, P and I are Jordan all-derivable points, where P is the standard non-trivial idempotent. In [33], Zhao and Zhu prove that every element in the algebra of all n × n upper triangular matrices over the complex field C is a Jordan all-derivable point. In Section 2, we give some general characterizations of Jordan derivable mappings, which will be used to determine Jordan all-derivable points for some general bimodules. Let A be a unital algebra and N be the set of non-negative integers. A sequence of mappings {d i } i∈N ∈ L(A) with d 0 = I A is called a higher derivation if d n (AB) = i+j=n d i (A)d j (B) for all A, B ∈ A; it is called a Jordan higher derivation if d n (AB + BA) = i+j=n (d i (A)d j (B) + d i (B)d j (A)) for all A, B ∈ A. With the development of derivations, the study of higher and Jordan higher derivations has attracted much attention as an active subject of research in operator algebras, and the local action problem ranks among in the list. A sequence of mappingsJor...

Abstract. Let A be a unital algebra, δ be a linear mapping from A into itself and m, n be fixed integers. We call δ an (m, n)-derivable mapping at Z, if mδpABq`nδpBAq " mδpAqB`mAδpBq`nδpBqA`nBδpAq for all A, B P A with AB " Z. In this paper, (m, n)-derivable mappings at 0 (resp. IA ' 0, I) on generalized matrix algebras are characterized. We also study (m, n)-derivable mappings at 0 on CSL algebras. We reveal the relationship between this kind of mappings with Lie derivations, Jordan derivations and derivations. IntroductionLet R be a unital ring and A be a unital R-algebra. Let δ be a linear mapping from A into itself. We call δ a derivation if δpABq " δpAqBÀ δpBq for all A, B P A. We call δ a Jordan derivation if δpAB`BAq " δpAqB`AδpBq`δpBqA`BδpAq for all A, B P A. δ is called a Lie derivation if δprA, Bsq " rδpAq, Bs`rA, δpBqs for all A, B P A, where rA, Bs " AB´BA. The questions of characterizing Jordan derivations and Lie derivations have received considerable attention from several authors, who revealed the relationship between Jordan derivations, derivations as well as Lie derivations (for example, [1,5,6,8,12] and the references therein).Let m, n be fixed integers. In [21], Vukman defined a new type of Jordan derivations, named pm, nq-Jordan derivation, that is, an additive mapping η from a ring R into itself such that pm`nqηpA 2 q " 2mηpAqA`2nAηpAq for every A P R. He proved that each pm, nq-Jordan derivation of a prime ring is a derivation. Motivated by this, we define a new type of derivations, named pm, nq-derivation. An pm, nq-derivation is a linear mapping δ from A into itself such that mδpABq`nδpBAq " mδpAqB`mAδpBq`nδpBqA`nBδpAq 2010 Mathematics Subject Classification: Primary 47L35; Secondary 16W25. Key words and phrases: CSL algebra, derivation, generalized matrix algebra, pm, nqderivation. for all A, B P A. Obviously, every p1, 1q-derivation is a Jordan derivation, each p1,´1q-derivation is a Lie derivation, p1, 0q-derivations and (0, 1)-derivations are derivations.Recently, there have been a number of papers on the study of conditions under which derivations on algebras can be completely determined by their action on some subsets of elements. Let δ be a linear mapping from A into itself and Z be in A. δ is called derivable at Z, if δpABq " δpAqB`AδpBq for all A, B P A with AB " Z; δ is called Jordan derivable at Z, if δpAB`BAq " δpAqB`AδpBq`δpBqA`BδpAq for all A, B P A with AB " Z; δ is called Lie derivable at Z, if δprA, Bsq " rδpAq, Bs`rA, δpBqs for all A, B P A with AB " Z. It is natural and interesting to ask whether or not a linear mapping is a derivation (Jordan derivation, or Lie derivation) if it is derivable (Jordan derivable, Lie derivable) only at one given point. An and Hou [2] investigated derivable mappings at 0, P, and I on triangular rings, where P is some fixed non-trivial idempotent. Let X be a Banach space, Lu and Jing [16] studied Lie derivable mappings at 0 and P on BpXq, where P is a fixed nontrivial idempotent. In [23], Zhao and Zhu characterized Jordan derivable mapping...

Let A be a unital algebra over a number field K. A linear mapping δ from A into itself is called a weak (m,n,l )-Jordan centralizer if (m + n + l)δ(A 2 ) − mδ(A)A − nAδ(A) − lAδ(I)A ∈ KI for every A ∈ A, where m ≥ 0, n ≥ 0, l ≥ 0 are fixed integers with m+n+l = 0. In this paper, we study weak (m,n,l )-Jordan centralizer on generalized matrix algebras and some reflexive algebras algL, where L is a CSL or satisfies ∨{L : L ∈ J (L)} = X or ∧{L − : L ∈ J (L)} = (0), and prove that each weak (m,n,l )-Jordan centralizer of these algebras is a centralizer when m + l ≥ 1 and n + l ≥ 1.

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