Approximate extension of partial ε-characters of abelian groups to characters with application to integral point lattices

Mačaj, Martin; Zlatoš, Pavol

doi:10.1016/s0019-3577(05)80026-6

Cited by 8 publications

(12 citation statements)

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“…Characters of LCA groups are globally stable-as LCA groups are amenable (see e g Pier [16]), this is a direct consequence of the following more general result proved in Cabello Sánchez [2] and (independently but later on) also in Mačaj-Zlatoš [14], slightly improving a special case of a theorem due to Kazhdan [12]. For convenience' sake we use the angular invariant metric…”

Section: Stability Of Characters Of Lca Groupsmentioning

confidence: 86%

“…Then the characters of discrete abelian groups are stable on finite sets. This is a direct consequence of the following more specific result proved in Mačaj, Zlatoš [14]. The number of elements of a finite set S ⊆ G is denoted by # S and S n denotes the subset of G generated in at most n steps from the elements of S and S −1 , i e,…”

Section: Stability Of Characters Of Lca Groupsmentioning

confidence: 87%

“…The proof in Mačaj-Zlatoš [14] is based on Theorem B and a nonstandard-analysis result by Gordon [7] (cf also Gordon [8]) implying that for each character g : X → T of a countable subgroup X of an ultraproduct G = i∈N G i /D of a system of abelian groups G i there exists an internal character γ :…”

Section: Stability Of Characters Of Lca Groupsmentioning

confidence: 99%

“…As we shall see, within the latter framework it is natural to deal not just with everywhere defined mappings G → H but also with partial mappings X → H for various subsets X ⊆ G. Using two older known results as well as a more recent one (marked as Theorems A, B and C) we will establish a kind of stability result by proving that characters of any LCA group are stable on compacts. This is more in spirit of the Pontryagin-van Kampen duality theory (where the duals of LCA groups are endowed with the compact-open topology) than the global stability of characters of LCA groups proved in Cabello Sánchez [2] and Mačaj-Zlatoš [14].…”

Section: Introductionmentioning

confidence: 99%

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Stability of group homomorphisms in the compact-open topology

Zlatoš¹

2010

JLA

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Abstract:We introduce the notion of stability of continuous homomorphisms of topological groups with respect to the compact-open topology on the space of all continuous functions between them and prove that the characters of any locally compact abelian group are stable in this sense. Using this notion we also generalize a global stability result on continuous homomorphisms between compact groups to a local version for continuous homomorphisms from any locally compact group to an arbitrary topological group and show the necessity of the assumptions of the theorem through some counterexamples. Finally, we separately discuss the special cases of metrizable and discrete groups, respectively. As an application of the latter we prove a purely algebraic result on extendability of finite partial functions to group homomorphisms.2000 Mathematics Subject Classification 22D12, 22B99 (primary); 22D10, 54H11, 54J05 (secondary)

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Section: Stability Of Characters Of Lca Groupsmentioning

confidence: 86%

Section: Stability Of Characters Of Lca Groupsmentioning

confidence: 87%

Section: Stability Of Characters Of Lca Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability of group homomorphisms in the compact-open topology

Zlatoš¹

2010

JLA

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“…As a consequence, Pontryaginvan Kampen duality is eliminated from the proof. Additionally, the passage from stability of characters to stability of dual lattices in [15] naturally led to a formulation in terms of the pair of mutually dual norms x 1 = |x 1 | + . .…”

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confidence: 99%

A uniform stability principle for dual lattices

Vodička

Zlatoš

2019

JLA

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We prove a highly uniform stability or "almost-near" theorem for dual lattices of lattices L ⊆ R n . More precisely, we show that, for a vector x from the linear span of a lattice L ⊆ R n , subject to λ 1 (L) ≥ λ > 0, to be ε-close to some vector from the dual lattice L ′ of L, it is enough that the inner products u x are δ-close (with δ < 1/3) to some integers for all vectors u ∈ L satisfying u ≤ r, where r > 0 depends on n, λ, δ and ε, only. This generalizes an analogous result proved for integral lattices in [15]. The proof is nonconstructive, using the ultraproduct construction and a slight portion of nonstandard analysis.

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Stability of Systems of General Functional Equations in the Compact-Open Topology

Zlatoš

2017

Developments in Functional Equations and Related Topics

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We introduce a fairly general concept of functional equation for k-tuples of functions f 1 , . . . , f k : X → Y between arbitrary sets. The homomorphy equations for mappings between groups and other algebraic systems, as well as various types of functional equations and recursion formulas occurring in mathematical analysis or combinatorics, respectively, become special cases (of systems) of such equations. Assuming that X is a locally compact and Y is a completely regular topological space, we show that systems of such functional equations, with parameters satisfying rather a modest continuity condition, are stable in the following intuitive sense: Every k-tuple of "sufficiently continuous," "reasonably bounded" functions X → Y satisfying the given system with a "sufficient precision" on a "big enough" compact set is already "arbitrarily close" on an "arbitrarily big" compact set to a k-tuple of continuous functions solving the system. The result is derived as a consequence of certain intuitively appealing "almost-near" principle using the relation of infinitesimal nearness formulated in terms of nonstandard analysis.2010 Mathematics Subject Classification. Primary 39B82;Secondary 39B72, 54D45, 54E15, 54J05. 1 2 P. ZLATOŠ (like, e.g., the sine and cosine addition formulas) or various recursion formulas occurring in combinatorics become just special cases (of systems) of such equations.Assuming that X is a locally compact and Y is a completely regular (i.e., uniformizable) topological space, we will show that systems of such functional equations, with functional parameters satisfying rather a modest continuity condition, are stable in the following intuitive sense, which will be made precise in the final Section 4 (Theorems 4.4, 4.9): Every k-tuple of "sufficiently continuous," "reasonably bounded" functions X → Y satisfying the given system with a "sufficient precision" on a "big enough" compact set is already "arbitrarily close" on an "arbitrarily big" compact set to a k-tuple of continuous functions solving the system. The result is a generalization comprising several former results by the author and his collaborators [SlZ], [SpZ], [Z1], [Z2], [Z3]. It is derived as a consequence of certain intuitively appealing stability or "almost-near" principle (in the sense of [An], [BB]) using the relation of infinitesimal nearness formulated in terms of nonstandard analysis in Section 3 (Theorem 3.1, Corollary 3.2), generalizing a more specific principle of this kind from [SlZ].

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Approximate extension of partial ε-characters of abelian groups to characters with application to integral point lattices

Cited by 8 publications

References 10 publications

Stability of group homomorphisms in the compact-open topology

Stability of group homomorphisms in the compact-open topology

A uniform stability principle for dual lattices

Stability of Systems of General Functional Equations in the Compact-Open Topology

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