Functional Identities

Brešar, Matej; Chebotar, Mikhail A.; Martindale, Wallace S.

doi:10.1007/978-3-7643-7796-0

Cited by 197 publications

(105 citation statements)

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“…We now consider the possible forms of this extension, using Theorem 1 from [5], which is to say Theorem 1.2 from the introduction. We note in particular that if Φ : span(MES Y,Y ) → span(MES Y,Y ) is invertible, then it must be of the form (1) or (2).…”

Section: The Preservers Of Mesmentioning

confidence: 99%

On the preservers of maximally entangled states

Großmann

Woerdeman

2019

Linear Algebra and its Applications

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Section: The Preservers Of Mesmentioning

confidence: 99%

On the preservers of maximally entangled states

Großmann

Woerdeman

2019

Linear Algebra and its Applications

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“…The condition that we will require is that the target Lie algebra is a 3-free subset of an associative unital algebra. To avoid making this section too lengthy, we only refer to the book [6] for the definition and basic properties of such sets, and give the proof of the next lemma without a detailed explanation of the results that are used. If the range of ϕ is a 3-free subset of A, then there exist λ ∈ Z, the center of A, and a skew-symmetric bilinear map ν :…”

Section: Commutativity Preserving Mapsmentioning

confidence: 99%

“…Proof. Since the range of ϕ is 3-free it follows from [6,Theorem 4.13] that there exist elements λ, λ ′ ∈ Z and maps µ, Since char(F) = 2 it follows from [6,Lemma 4.4] that µ = 0.…”

Section: Commutativity Preserving Mapsmentioning

confidence: 99%

Zero product determined Lie algebras

Brešar

Guo

Liu

et al. 2018

European Journal of Mathematics

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A Lie algebra L over a field F is said to be zero product determined (zpd) if every bilinear map f : L × L → F with the property that f (x, y) = 0 whenever x and y commute is a coboundary. The main goal of the paper is to determine whether or not some important Lie algebras are zpd. We show that the Galilei Lie algebra sl2 ⋉ V , where V is a simple sl2-module, is zpd if and only if dim V = 2 or dim V is odd. The class of zpd Lie algebras also includes the quantum torus Lie algebras Lq and L + q , the untwisted affine Lie algebras, the Heisenberg Lie algebras, and all Lie algebras of dimension at most 3, while the class of non-zpd Lie algebras includes the (4-dimensional) aging Lie algebra age(1) and all Lie algebras of dimension more than 3 in which only linearly dependent elements commute. We also give some evidence of the usefulness of the concept of a zpd Lie algebra by using it in the study of commutativity preserving linear maps.

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“…We begin by recalling a result from the theory of commuting mappings. It involves the notion of extended centroid of a prime ring R. We refer the reader to [3] for a full account on these concepts. In general, the extended centroid is a field that contains the centre of R as a subring.…”

Section: Proofsmentioning

confidence: 99%

Commuting holomorphic maps on the spectral unit ball

Costara

2009

Bulletin of the London Mathematical Society

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We prove that if F is a holomorphic map from the open spectral unit ball of a primitive Banach algebra into itself satisfying F (0) = 0, F (0) = I and F (x)x = xF (x) for every x, then F is the identity map. Using this, we prove that if A is a semisimple Banach algebra and B is a primitive Banach algebra, then any unital spectral isometry from A onto B which locally preserves commutativity is a Jordan morphism. The same is true when A and B are both assumed to be von Neumann algebras.

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Functional Identities

Cited by 197 publications

References 0 publications

On the preservers of maximally entangled states

On the preservers of maximally entangled states

Zero product determined Lie algebras

Commuting holomorphic maps on the spectral unit ball

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