Algebraic characterizations of locally compact groups

This paper is devoted to the study of orthogonality and disjointness preserving linear maps between Fourier and Fourier-Stieltjes algebras of locally compact groups. We show that a linear bijection Ψ :) between two Fourier algebras (resp. Fourier-Stieltjes algebras) of locally compact groups will induce a topological group isomorphism between G 1 and G 2 , provided that Ψ preserves both disjointness and some kind of orthogonality. This improves earlier results of J. J. Font and M. S. Monfared, where amenability of the groups or continuity of the linear maps are assumed. We also study the structure of bounded and unbounded disjointness preserving linear functionals of Fourier algebras. In the development, general results about disjointness and orthogonality preserving linear maps between C*-algebras, W*-algebras and their preduals are obtained.

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“…The same is true for the group algebras of locally compact groups (see, e.g., [22,23,20,45]). Let G be a locally compact group.…”

Section: Introductionmentioning

confidence: 84%

“…In this setting, f, g in A(G i ) are orthogonal as normal functionals of V N (G i ) ∼ = L ∞ ( G i ) if and only if f g = 0 in L 1 ( G i ), i.e., f , g have disjoint cozero sets in G i . The condition (23) amounts to say that Ψ is orthogonality preserving.…”

Section: Disjointness and Orthogonality Together Determine Topological Group Structuresmentioning

confidence: 99%

Orthogonality and disjointness preserving linear maps between Fourier and Fourier–Stieltjes algebras of locally compact groups

Lau

Wong

2013

Journal of Functional Analysis

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“…As a result, some topological links between underlying spaces are deduced, and weighted composition type representations for separating maps are obtained. In recent years, considerable attention has been given to separating maps; see for example [1], [2], and [4] on Banach lattices, [3] and [6] on spaces of continuous functions, and [5] on group algebras of locally compact Abelian groups.…”

Section: Introductionmentioning

confidence: 99%

Separating maps on weighted function algebras on topological groups

Maghsoudi

Nasr-Isfahani

2011

Mathematica Slovaca

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Let G 1 and G 2 be locally compact groups and let ω 1 and ω 2 be weight functions on G 1 and G 2, respectively. For i = 1, 2, let also C 0(G i, 1/ω i) be the algebra of all continuous complex-valued functions f on G i such that f/ω i vanish at infinity, and let H: C 0(G 1, 1/ω 1) → C 0(G 2, 1/ω 2) be a separating map; that is, a linear map such that H(f)H(g) = 0 for all f, g ∈ C 0(G 1, 1/ω 1) with fg = 0. In this paper, we study conditions under which H can be represented as a weighted composition map; i.e., H(f) = φ(f ℴ h) for all f ∈ C 0(G 1, 1/ω 1), where φ: G 2 → ℂ is a non-vanishing continuous function and h: G 2 → G 1 is a topological isomorphism. Finally, we offer a statement equivalent to that h is also a group homomorphism.

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“…Subsequently, several results were gained on locally compact groups. For example, Font and Hernandez studied separating maps between the algebra of bounded continuous maps on locally compact groups in [15,17]. In [16], they used the separating maps between Fourier algebras of two locally compact abelian groups to study the separating maps on group algebras of locally compact abelian groups.…”

mentioning

confidence: 99%

Separating maps between commutative Banach algebras

Alaghmandan,

Nasr-Isfahani,

Nemati

2011

Preprint

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Let A and B be Banach algebras. A linear map T : A → B is called separating or disjointness preserving if ab = 0 implies T a T b = 0 for all a, b ∈ A. In this paper, we study a new class of regular Tauberian algebras and prove that some well-known Banach algebras in harmonic analysis belong to this class. We show that a bijective separating map between these algebras turns out to be continuous and the maximal ideal spaces of underlying algebras are homeomorphic. By imposing extra conditions on these algebras, we find a more thorough characterization of separating maps. The existence of a bijective separating map also leads to the existence of an algebraic isomorphism in some cases.

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Algebraic characterizations of locally compact groups

Cited by 5 publications

References 18 publications

Orthogonality and disjointness preserving linear maps between Fourier and Fourier–Stieltjes algebras of locally compact groups

Orthogonality and disjointness preserving linear maps between Fourier and Fourier–Stieltjes algebras of locally compact groups

Separating maps on weighted function algebras on topological groups

Separating maps between commutative Banach algebras

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