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Automatic continuity of operators commuting with translations

100%
Alaminos J., Extremera J., Villena A.
Studia Mathematica
|
2006
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tom 173
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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.
2
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Uniqueness of the topology on L¹(G)

100%
Extremera J., Mena J., Villena A.
Studia Mathematica
|
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.
3
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Maps preserving zero products

100%
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.
4
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Stability of commuting maps and Lie maps

100%
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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