Automorphisms and derivations of upper triangular matrix rings

Jøndrup, S.

doi:10.1016/0024-3795(93)00255-x

Cited by 68 publications

(27 citation statements)

References 6 publications

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“…In the past six decades, many authors studied linear preserver problems (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]) which are concerning with characterizing maps on matrix spaces that preserve some property, set or relation. As pointed out in [5,7], one of techniques that have been successful in solving linear preserver problems is to reducing a new problem to a known one, while the latter is often a preserver of idempotence.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Linear preservers of idempotence on triangular matrix spaces over any field

Xu¹,

Cao²,

Tang³

2007

imf

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Let F be any field and let T n (F) be the n × n upper triangular matrix space over F. We denote the set of all n × n upper triangular idempotent matrices over F by P n (F). A map ϕ on T n (F) is called a preserver of idempotence if ϕ(P n (F)) ⊂ P n (F); and a strong preserver of idempotence if ϕ(P n (F)) = P n (F). In this paper, we characterize the bijective linear preservers of idempotence on T n (F). Further, the strong linear preservers of idempotence on T n (F) are characterized. Mathematics Subject Classifications: 15A04; 15A03

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Linear preservers of idempotence on triangular matrix spaces over any field

Xu¹,

Cao²,

Tang³

2007

imf

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“…To do this, we will proceed by induction. It should be mentioned that the techniques used here are similar to those in the proof of [17,Theorem 2], which characterizes the derivations of matrix algebras. The conclusion holds clearly for n = 1, since every derivation of A is inner.…”

Section: Theorem 24 Let a Be A Unital C * -Algebra Suppose That Evementioning

confidence: 97%

Derivations on the algebra of operators in hilbert C*-modules

Han

Tang

2012

Acta. Math. Sin.-English Ser.

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Let M be a full Hilbert C * -module over a C * -algebra A, and let End * A (M) be the algebra of adjointable operators on M. We show that if A is unital and commutative, then every derivation of End * A (M) is an inner derivation, and that if A is σ-unital and commutative, then innerness of derivations on "compact" operators completely decides innerness of derivations on End * A (M). If A is unital (no commutativity is assumed) such that every derivation of A is inner, then it is proved that every derivation of End * A (L n (A)) is also inner, where L n (A) denotes the direct sum of n copies of A. In addition, in case A is unital, commutative and there exist x 0 , y 0 ∈ M such that x 0 , y 0 = 1, we characterize the linear A-module homomorphisms on End * A (M) which behave like derivations when acting on zero products.

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“…Rota and Stanley developed incidence algebras as the fundamental structures of enumerative combinatorial theory and allied areas of arithmetic function theory (see [15]). Motivated by the results of Stanley [17], automorphisms and other algebraic mappings of incidence algebras have been extensively studied (see [1,5,9,10,12,13,14,16] and the references therein). Baclawski [1] studied the automorphisms and derivations of incidence algebras I(X, R) when X is a locally finite partially ordered set.…”

Section: Introductionmentioning

confidence: 99%

Jordan derivations of incidence algebras

Xiao¹

2015

Rocky Mountain J. Math.

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Abstract. Let R be a commutative ring with identity, I(X, R) be the incidence algebra of a locally finite pre-ordered set X. In this note, we characterise the derivations of I(X, R) and prove that every Jordan derivation of I(X, R) is a derivation provided that R is 2-torsion free. IntroductionLet R be a commutative ring with identity, A be an algebra over R. An Rlinear mapping D : A → A is called a derivation if D(xy) = D(x)y + xD(y) for all x, y ∈ A, and is called a Jordan derivation iffor all x ∈ A. There has been a great interest in the study of Jordan derivations of various algebras in the last decades. The standard problem is to find out whether a Jordan derivation degenerate to a derivation. Jacobson and Rickart [8] proved that every Jordan derivation of the full matrix algebra over a 2-torsion free unital ring is a derivation by relating the problem to the decomposition of Jordan homomorphisms. In [7], Herstein showed that every Jordan derivation from a 2-torsion free prime ring into itself is also a derivation. These results have been extended to different rings and algebras in various directions (see [2,3,6,11,20] and the references therein). We would like to refer the reader to Brešar's paper [4] for a comprehensive and more detailed understanding of this topic. We now recall the definition of incidence algebras. Let (X, ) be a locally finite pre-ordered set. This means is a reflexive and transitive binary relation on the set X, and for any x y in X there are only finitely many elements z satisfying x z y. The incidence algebra I(X, R) of X over R is defined on the setwith algebraic operation given byfor all f, g ∈ I(X, R), r ∈ R and x, y, z ∈ X. The product (f g) is usually called convolution in function theory. It would be helpful to point out that the full matrix 2010 Mathematics Subject Classification. 16W10, 16W25, 47L35.

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Automorphisms and derivations of upper triangular matrix rings

Cited by 68 publications

References 6 publications

Linear preservers of idempotence on triangular matrix spaces over any field

Linear preservers of idempotence on triangular matrix spaces over any field

Derivations on the algebra of operators in hilbert C*-modules

Jordan derivations of incidence algebras

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