Separating maps on weighted function algebras on topological groups

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
Saeid Maghsoudi ◽  
Rasoul Nasr-Isfahani
Keyword(s):  
Weight Functions ◽  
Compact Groups ◽  
Topological Isomorphism ◽  
Topological Groups ◽  
Locally Compact ◽  
Weighted Function ◽  
Function Algebras ◽  
Weighted Composition ◽  
Continuous Complex ◽  
Complex Valued

AbstractLet 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.

Compact left ideal groups in semigroup compactification of locally compact groups

1993 ◽  
Vol 113 (3) ◽  
pp. 507-517 ◽  
Author(s):  
J. W. Baker ◽  
A. T. Lau
Keyword(s):  
Minimal Ideal ◽  
Locally Compact Group ◽  
Maximal Subgroups ◽  
Compact Groups ◽  
Locally Compact ◽  
Semigroup Compactification ◽  
Continuous Complex ◽  
Complex Valued ◽  
Uniformly Continuous ◽  
Right Topological Semigroup

Let G be a locally compact group and let UG denote the spectrum of the C*-algebra LUC(G) of bounded left uniformly continuous complex-valued functions on G, with the Gelfand topology. Then there is a multiplication on UG extending the multiplication on G (when naturally embedded in UG) such that UG is a semigroup and for each x ∈ UG, the map y ↦ yx from UG into UG is continuous, i.e. UG is a compact right topological semigroup. Consequently UG has a unique minimal ideal K which is the union of minimal (closed) left ideals UG. Furthermore K is the union of the set of maximal subgroups of K (see [3], theorem 3·ll).

A decomposition theorem for locally compact groups

2019 ◽  
Vol 26 (1) ◽  
pp. 29-33
Author(s):  
Sanjib Basu ◽  
Krishnendu Dutta
Keyword(s):  
Compact Group ◽  
Lebesgue Point ◽  
Decomposition Theorem ◽  
Locally Compact Group ◽  
The Other ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Interesting Connection

Abstract We prove that, under certain restrictions, every locally compact group equipped with a nonzero, σ-finite, regular left Haar measure can be decomposed into two small sets, one of which is small in the sense of measure and the other is small in the sense of category, and all such decompositions originate from a generalised notion of a Lebesgue point. Incidentally, such class of topological groups for which this happens turns out to be metrisable. We also observe an interesting connection between Luzin sets in such spaces and decompositions of the above type.

Spectral synthesis on zero-dimensional locally compact Abelian groups

2019 ◽  
pp. 450-456
Author(s):  
Sergey S. Platonov
Keyword(s):  
Invariant Subspace ◽  
Linear Subspace ◽  
Spectral Synthesis ◽  
Linear Span ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Closed Linear Subspace ◽  
Compact Abelian Groups ◽  
Continuous Complex ◽  
Complex Valued

Let G be a zero-dimensional locally compact Abelian group whose elements are compact, C(G) the space of continuous complex-valued functions on the group G. A closed linear subspace H⊆ C(G) is called invariant subspace, if it is invariant with respect to translations τ_y ∶ f(x) ↦ f(x + y), y ∈ G. We prove that any invariant subspace H admits spectral synthesis, which means that H coincides with the closure of the linear span of all characters of the group G contained in H.

scholarly journals Locally compact products and coproducts in categories of topological groups

1977 ◽  
Vol 17 (3) ◽  
pp. 401-417 ◽  
Author(s):  
Karl Heinrich Hofmann ◽  
Sidney A. Morris
Keyword(s):  
Abelian Groups ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Torsion Free Group ◽  
Locally Compact Abelian Groups ◽  
Discrete Torsion ◽  
Compact Abelian Groups ◽  
Torsion Free

In the category of locally compact groups not all families of groups have a product. Precisely which families do have a product and a description of the product is a corollary of the main theorem proved here. In the category of locally compact abelian groups a family {Gj; j ∈ J} has a product if and only if all but a finite number of the Gj are of the form Kj × Dj, where Kj is a compact group and Dj is a discrete torsion free group. Dualizing identifies the families having coproducts in the category of locally compact abelian groups and so answers a question of Z. Semadeni.

scholarly journals LOCALLY COMPACT WREATH PRODUCTS

2018 ◽  
Vol 107 (1) ◽  
pp. 26-52 ◽  
Author(s):  
YVES CORNULIER
Keyword(s):  
Normal Subgroup ◽  
Wreath Product ◽  
Natural Extension ◽  
Wreath Products ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Intermediate Growth ◽  
Compactly Generated

Wreath products of nondiscrete locally compact groups are usually not locally compact groups, nor even topological groups. As a substitute introduce a natural extension of the wreath product construction to the setting of locally compact groups. Applying this construction, we disprove a conjecture of Trofimov, constructing compactly generated locally compact groups of intermediate growth without any open compact normal subgroup.

On multipliers of Fourier transforms

1972 ◽  
Vol 71 (1) ◽  
pp. 63-66 ◽  
Author(s):  
Louis Pigno
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Lebesgue Space ◽  
Fourier Transforms ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Continuous Complex ◽  
Complex Valued

In this paper G is a locally compact Abelian group, φ a complex-valued function defined on the dual Γ, Lp(G) (1 ≤ p ≤ ∞) the usual Lebesgue space of index p formed with respect to Haar measure, C(G) the set of all bounded continuous complex-valued functions on G, and C0(G) the set of all f ∈ C(G) which vanish at infinity.

Studies in harmonic analysis

1964 ◽  
Vol 60 (3) ◽  
pp. 465-516 ◽  
Author(s):  
N. Th. Varopoulos
Keyword(s):  
Harmonic Analysis ◽  
Complete Characterization ◽  
Locally Compact Groups ◽  
Topological Spaces ◽  
Compact Groups ◽  
Measure Algebra ◽  
Topological Groups ◽  
Inductive Limits ◽  
Locally Compact ◽  
Character Group

In this paper we shall be mainly concerned with the following three apparently widely differing questions.(a) What are the possible group topologies on an Abelian group that have a given, fixed continuous character group?In developing our theory, we are very strongly motivated by the duality theory of linear topological spaces and in particular by Mackey's theorem of that theory. This important result gives a complete characterization of all locally convex topologies on a linear space that have a given, fixed, separating dual space. The analogue of Mackey's theorem for groups, together with related results, is examined in sections 1 and 2 of part 2 of the paper.(b) What are the properties of topological groups that are denumerable inductive limits of locally compact groups? (See section 1 of part 1 of the paper for definitions.)Our aim here is to extend results known for locally compact groups to this larger class of groups. The topological study of these groups is carried out in section 3 of part 1 of the paper and the really deep results about their characters are proved in section 5 of part 3 of the paper, as applications of the theory developed in that part of the paper, which is a type of harmonic analysis for these groups.(c) What are the properties of certain algebras of measures of a locally compact group G, that strictly contain L1(G), and share most of the pleasing properties of L1(G), that is, they do not have any of the pathological features of the full measure algebra M(G) such as the Wiener–Pitt phenomenon or asymmetry?

scholarly journals Weighted function algebras on groups and semigroups

1986 ◽  
Vol 33 (2) ◽  
pp. 307-318 ◽  
Author(s):  
Heneri A. M. Dzinotyiweyi
Keyword(s):  
Continuous Functions ◽  
Almost Periodic ◽  
Locally Compact ◽  
Weighted Function ◽  
Weakly Almost Periodic ◽  
Function Algebras ◽  
Weighted Algebra ◽  
Necessary And Sufficient Criteria ◽  
Necessary And Sufficient ◽  
Locally Compact Topological Group

For a locally compact topological group admitting a weight function, we establish necessary and sufficient criteria for all the weighted continuous functions to be weakly almost periodic. Among other results, we show that weak almost periodicity of all ω-weighed continuous functions on a discrete semigroup S, can be very different drom the phenomenon of regularity of multiplication in the weighted algebra ℓ1 (S, w).

LOCALLY COMPACT GROUPS ADMITTING FAITHFUL STRONGLY CHAOTIC ACTIONS ON HAUSDORFF SPACES

2013 ◽  
Vol 23 (09) ◽  
pp. 1350158 ◽  
Author(s):  
FRIEDRICH MARTIN SCHNEIDER ◽  
SEBASTIAN KERKHOFF ◽  
MIKE BEHRISCH ◽  
STEFAN SIEGMUND
Keyword(s):  
Hausdorff Space ◽  
Geometric Characterization ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Continuous Action

In this paper we provide a geometric characterization of those locally compact Hausdorff topological groups which admit a faithful strongly chaotic continuous action on some Hausdorff space.

scholarly journals A Universal Property of the Takahashi Quasi-Dual

1972 ◽  
Vol 24 (3) ◽  
pp. 530-536 ◽  
Author(s):  
Detlev Poguntke
Keyword(s):  
Topological Group ◽  
Compact Group ◽  
Duality Theorem ◽  
Commutator Subgroup ◽  
Continuous Homomorphism ◽  
Locally Compact Group ◽  
Compact Groups ◽  
Almost Periodic ◽  
Topological Groups ◽  
Locally Compact

Topological group always means Hausdorff topological group, homomorphism (isomorphism) between topological groups always means continuous homomorphism (homeomorphic isomorphism). For a topological group G, the topological commutator subgroup (the closure of the algebraic commutator subgroup) is denoted by G’. For each locally compact group G, Takahashi has constructed a locally compact group GT (called the Takahashi quasi-dual) and a homomorphism G → GT such that GT is maximally almost periodic, and GT’ is compact. The category of all locally compact groups with these two properties is denoted by [TAK]. Takahashi's duality theorem states that G → GT is an isomorphism if G ∊ [TAK].

Sign in / Sign up

Export Citation Format

Share Document