On maps preserving Lie products equal to a rank-one nilpotent

Ginsburg, V; Julius, Hayden; Velásquez, Ricardo A.

doi:10.1016/j.laa.2020.02.007

Cited by 8 publications

(3 citation statements)

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“…Fixed product preserving mappings also arise naturally for other operations, such as the Lie product [a, b] = ab − ba and the Jordan product a • b = ab + ba, which are of principal interest. Some relevant problems for the Lie product can be found in [18,21,26,28]. For the Jordan product, see [8,13]; most notably, the authors in [8] obtained a complete description for maps preserving equal fixed Jordan products on M n (C).…”

Section: Introductionmentioning

confidence: 99%

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Fixed product preserving mappings on Banach algebras

Julius¹

2022

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In this paper, we describe linear maps between complex Banach algebras that preserve products equal to fixed elements. This generalizes some important special cases where the fixed elements are the zero or identity element. First we show that if such map preserves products equal to a finiterank operator, then it must also preserve the zero product. In several instances, this is enough to show that a product preserving map must be a scalar multiple of an algebra homomorphism. Second, we explore a more general problem concerning the existence of product preserving maps and the relationship between the fixed elements. Lastly, motivated by Kaplansky's problem on invertibility preservers, we show that maps preserving products equal to fixed invertible elements are either homomorphisms or antihomomorphisms multiplied on the left by a fixed element.

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Section: Introductionmentioning

confidence: 99%

“…For the Jordan product, see [8,13]; most notably, the authors in [8] obtained a complete description for maps preserving equal fixed Jordan products on M n (C). However, the Lie case can be pathological (such as in [18]). General approaches (even for M n (C)) for the Lie product are not known and seems to be a challenge.…”

Section: Introductionmentioning

confidence: 99%

Fixed product preserving mappings on Banach algebras

Julius¹

2022

Preprint

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“…When M n (C) is equipped with the Lie bracket [a, b] = ab − ba, the description slightly differs, but still reminds the description of commutativity preservers (i.e. zero product preservers with respect to [−, −]), see [14,15]. In the setting of C * -algebras the problem of studying linear mappings that are * -homomorphisms at a fixed point has been considered in [3].…”

Section: Introductionmentioning

confidence: 99%

Linear maps preserving products equal to primitive idempotents of an incidence algebra

Garcés¹,

Khrypchenko²

2022

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Let A, B be algebras and a ∈ A, b ∈ B a fixed pair of elements. We say that a map ϕ : A → B preserves products equal to a and b if for all a 1 , a 2 ∈ A the equality a 1 a 2 = a implies ϕ(a 1 )ϕ(a 2 ) = b. In this paper we study bijective linear maps ϕ : I(X, F ) → I(X, F ) preserving products equal to primitive idempotents of I(X, F ), where I(X, F ) is the incidence algebra of a finite connected poset X over a field F . We fully characterize the situation, when such a map ϕ exists, and whenever it does, ϕ is either an automorphism of I(X, F ) or the negative of an automorphism of I(X, F ).

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