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The range of a derivation on a Jordan-Banach algebra

100%
Brešar M., Villena A.
Studia Mathematica
|
2001
|
tom 146
|
nr 2
177-200
EN
The questions when a derivation on a Jordan-Banach algebra has quasi-nilpotent values, and when it has the range in the radical, are discussed.
2
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Maps preserving zero products

81%
Alaminos J., Brešar M., Extremera J., Villena A.
Studia Mathematica
|
2009
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tom 193
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nr 2
131-159
EN
A linear map T from a Banach algebra A into another B preserves zero products if T(a)T(b) = 0 whenever a,b ∈ A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T: A → B that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras. Our method involves continuous bilinear maps ϕ: A × A → X (for some Banach space X) with the property that ϕ(a,b) = 0 whenever a,b ∈ A are such that ab = 0. We prove that such a map necessarily satisfies ϕ(aμ,b) = ϕ(a,μ b) for all a,b ∈ A and for all μ from the closure with respect to the strong operator topology of the subalgebra of ℳ(A) (the multiplier algebra of A) generated by doubly power-bounded elements of ℳ(A). This method is also shown to be useful for characterizing derivations through the zero products.
3
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Lie triple ideals and Lie triple epimorphisms on Jordan and Jordan-Banach algebras

81%
Brešar M., Cabrera M., Fošner M., Villena A.
Studia Mathematica
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2005
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tom 169
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nr 3
207-228
EN
A linear subspace M of a Jordan algebra J is said to be a Lie triple ideal of J if [M,J,J] ⊆ M, where [·,·,·] denotes the associator. We show that every Lie triple ideal M of a nondegenerate Jordan algebra J is either contained in the center of J or contains the nonzero Lie triple ideal [U,J,J], where U is the ideal of J generated by [M,M,M]. Let H be a Jordan algebra, let J be a prime nondegenerate Jordan algebra with extended centroid C and unital central closure Ĵ, and let Φ: H → J be a Lie triple epimorphism (i.e. a linear surjection preserving associators). Assume that deg(J) ≥ 12. Then we show that there exist a homomorphism Ψ: H → Ĵ and a linear map τ: H → C satisfying τ([H,H,H]) = 0 such that either Φ = Ψ + τ or Φ = -Ψ + τ. Using the preceding results we show that the separating space of a Lie triple epimorphism between Jordan-Banach algebras H and J lies in the center modulo the radical of J.
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