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1
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Derivations with a hereditary domain, II

100%
Villena A.
Studia Mathematica
|
1998
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tom 130
|
nr 3
275-291
EN
The nilpotency of the separating subspace of an everywhere defined derivation on a Banach algebra is an intriguing question which remains still unsolved, even for commutative Banach algebras. On the other hand, closability of partially defined derivations on Banach algebras is a fundamental problem motivated by the study of time evolution of quantum systems. We show that the separating subspace S(D) of a Jordan derivation defined on a subalgebra B of a complex Banach algebra A satisfies $B[B ∩ S(D)]B ⊂ Rad_B(A)$ provided that BAB ⊂ A and $dim(Rad_J(A) ∩ ⋂_{n=1}^∞ B^n) < ∞$, where $Rad_J(A)$ and $Rad_B(A)$ denote the Jacobson and the Baer radicals of A respectively. From this we deduce the closability of partially defined derivations on complex semiprime Banach algebras with appropriate domains. The result applies to several relevant classes of algebras.
2
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The range of a derivation on a Jordan-Banach algebra

64%
Brešar M., Villena A.
Studia Mathematica
|
2001
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tom 146
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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.
3
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Automatic continuity of operators commuting with translations

51%
Alaminos J., Extremera J., Villena A.
Studia Mathematica
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2006
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tom 173
|
nr 3
259-293
EN
Let $τ_{X}$ and $τ_{Y}$ be representations of a topological group G on Banach spaces X and Y, respectively. We investigate the continuity of the linear operators Φ: X → Y with the property that $Φ ∘ τ_{X}(t) = τ_{Y}(t) ∘ Φ $ for each t ∈ G in terms of the invariant vectors in Y and the automatic continuity of the invariant linear functionals on X.
4
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Uniqueness of the topology on L¹(G)

51%
Extremera J., Mena J., Villena A.
Studia Mathematica
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2002
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tom 150
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nr 2
163-173
EN
Let G be a locally compact abelian group and let X be a translation invariant linear subspace of L¹(G). If G is noncompact, then there is at most one Banach space topology on X that makes translations on X continuous. In fact, the Banach space topology on X is determined just by a single nontrivial translation in the case where the dual group Ĝ is connected. For G compact we show that the problem of determining a Banach space topology on X by considering translation operators on X is closely related to the classical problem of determining whether or not there is a discontinuous translation invariant linear functional on X. As a matter of fact L¹(G) does not carry a unique Banach space topology that makes translations continuous, but translations almost determine the Banach space topology on X. Moreover, if G is connected and compact and 1 < p < ∞, then $L^{p}(G)$ carries a unique Banach space topology that makes translations continuous.
5
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Maps preserving zero products

51%
Alaminos J., Brešar M., Extremera J., Villena A.
Studia Mathematica
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2009
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tom 193
|
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.
6
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Lie triple ideals and Lie triple epimorphisms on Jordan and Jordan-Banach algebras

51%
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.
7
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Norm attaining bilinear forms on C*-algebras

51%
Alaminos J., Payá R., Villena A.
Studia Mathematica
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2003
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tom 157
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nr 1
47-56
EN
We give a sufficient condition on a C*-algebra to ensure that every weakly compact operator into an arbitrary Banach space can be approximated by norm attaining operators and that every continuous bilinear form can be approximated by norm attaining bilinear forms. Moreover we prove that the class of C*-algebras satisfying this condition includes the group C*-algebras of compact groups.
8
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Stability of commuting maps and Lie maps

51%
Alaminos J., Extremera J., Špenko Š., Villena A.
Studia Mathematica
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2012
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tom 213
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nr 1
25-48
EN
Let A be an ultraprime Banach algebra. We prove that each approximately commuting continuous linear (or quadratic) map on A is near an actual commuting continuous linear (resp. quadratic) map on A. Furthermore, we use this analysis to study how close are approximate Lie isomorphisms and approximate Lie derivations to actual Lie isomorphisms and Lie derivations, respectively.
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