On maps preserving zeros of the polynomial xy−yx*

Chebotar, Mikhail A.; Fong, Y.; Lee, Pjek-Hwee

doi:10.1016/j.laa.2005.06.015

Cited by 27 publications

(18 citation statements)

References 37 publications

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“…Šemrl [14] showed that every Jordan * -derivation Let R be a * -ring. Recall that a map Φ : R → R is skew commutativity preserving if [Φ(A), Φ(B)] * = 0 whenever [A, B] * = 0 for all A, B ∈ R. The problem of characterizing linear or additive bijective maps preserving skew commutativity had been investigated on various algebras (see [2,3]). More specially, we say that a map Φ : R → R is strong skew commutativity…”

Section: Introductionmentioning

confidence: 99%

Strong skew commutativity preserving maps on rings with involution

Chen

2016

Acta. Math. Sin.-English Ser.

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Let R be a unital * -ring with the unit I. Assume that R contains a symmetric idempotent P which satisfies ARP = 0 implies A = 0 and AR(I − P ) = 0 implies A = 0. In this paper, it is shown that a surjective map Φ : R → R is strong skew commutativity preserving (that is, satisfiesis the symmetric center of R. As applications, the strong skew commutativity preserving maps on unital prime * -rings and von Neumann algebras with no central summands of type I 1 are characterized.

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Section: Introductionmentioning

confidence: 99%

Strong skew commutativity preserving maps on rings with involution

Chen

2016

Acta. Math. Sin.-English Ser.

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“…The problem of characterizing linear (or additive) bijective maps preserving skew commutativity had been studied intensively on various algebras (see [5,6] and the references therein). More specially, we say that a map Φ : R → R is strong skew commutativity preserving (briefly, SSCP) if [Φ(A), Φ(B)] * = [A, B] * for all A, B ∈ R. SSCP maps are also called strong skew Lie product preserving maps in [7].…”

Section: Introductionmentioning

confidence: 99%

Strong skew commutativity preserving maps on von Neumann algebras

Hou

2013

Journal of Mathematical Analysis and Applications

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Let ${\mathcal M}$ be a von Neumann algebra without central summands of type $I_1$. Assume that $\Phi:{\mathcal M}\rightarrow {\mathcal M}$ is a surjective map. It is shown that $\Phi$ is strong skew commutativity preserving (that is, satisfies $\Phi(A)\Phi(B)-\Phi(B)\Phi(A)^*=AB-BA^*$ for all $A,B\in{\mathcal M}$) if and only if there exists some self-adjoint element $Z$ in the center of ${\mathcal M}$ with $Z^2=I$ such that $\Phi(A)=ZA$ for all $A\in{\mathcal M}$. The strong skew commutativity preserving maps on prime involution rings and prime involution algebras are also characterized.Comment: 16 page

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“…Many authors have investigated preserver problems on rings with involution [1][2][3][4][5][6][7]. In general rings, a novel technique of functional identities proved to be indispensable [1]; this was also successfully tried in matrix algebras [3].…”

Section: Introductionmentioning

confidence: 99%