scholarly journals A Theorem on Algebras of Measures on Topological Groups

1959 ◽  
Vol 11 (4) ◽  
pp. 195-206 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Topological Groups ◽  
Finite Sets ◽  
Locally Compact ◽  
Borel Measures ◽  
Bounded Complex ◽  
Image Position

Let G be a locally compact Abelian group, and the set of bounded complex (regular countably-additive Borel) measures on G. It is well known that becomes a Banach space if the norm is defined bythe supremum being over all finite sets of disjoint Borel subsets of G.

scholarly journals On uniformly distributed sequences of integers and Poincaré recurrence III

2003 ◽  
Vol 68 (2) ◽  
pp. 345-350
Author(s):  
R. Nair
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Probability Space ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Bohr Compactification ◽  
Poincare Recurrence ◽  
Uniformly Distributed Sequences ◽  
Image Position ◽  
Measure Preserving Transformations

Let S be a semigroup contained in a locally compact Abelian group G. Let Ĝ denote the Bohr compactification of G. We say that a sequence contained in S is Hartman uniform distributed on G iffor any character χ in Ĝ. Suppose that (Tg)g∈s is a semigroup of measurable measure preserving transformations of a probability space (X, β, μ) and B is an element of the σ-algebra β of positive μ measure. For a map T: X → X and a set A ⊆ X let T−1A denote {x ∈ X: Tx ∈ A}. In an earlier paper, the author showed that if k is Hartman uniform distributed thenIn this paper we show that ≥ cannot be replaced by =. A more detailed discussion of this situation ensues.

scholarly journals Multipliers for Amalgams and the Algebra S0(G)

1987 ◽  
Vol 39 (1) ◽  
pp. 123-148 ◽  
Author(s):  
Maria L. Torres De Squire
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
The Difference ◽  
Image Position

Throughout the whole paper G will be a locally compact abelian group with Haar measure m and dual group Ĝ. The difference of two sets A and B will be denoted by A ∼ B, i.e.,For a function f on G and s ∊ G, the functions f′ and fs will be defined by

Banach space valued weak second order stochastic processes

2021 ◽  
pp. 71-96
Author(s):  
Yûichirô Kakihara
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Abelian Group ◽  
Stochastic Processes ◽  
Compact Abelian Group ◽  
Covariance Function ◽  
Second Order ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Content Type

Banach space valued stochastic processes of weak second order on a locally compact abelian group G G is considered. These processes are recognized as operator valued processes on G G . More fully, letting U \mathfrak {U} be a Banach space and H \mathfrak {H} a Hilbert space, we study B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued processes. Since B ( U , H ) B(\mathfrak {U},\mathfrak {H}) has a B ( U , U ∗ ) B(\mathfrak {U},\mathfrak {U}^*) -valued gramian, every B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued process has a B ( U , U ∗ ) B(\mathfrak {U},\mathfrak {U}^*) -valued covariance function. Using this property we can define operator stationarity, operator harmonizability and operator V V -boundedness for B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued processes, in addition to scalar ones. Interrelations among these processes are obtained together with the operator stationary dilation.

scholarly journals Supports and singular supports of pseudomeasures

1966 ◽  
Vol 6 (1) ◽  
pp. 65-75 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Closed Subset ◽  
Sufficient Conditions ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Singular Support ◽  
Image Position

SummaryLet G denote a Hausdorff locally compact Abelian group which is nondiscrete and second countable. The main results (Theorems (2.2) and (2.3)) assert that, for any closed subset E of G there exists a pseudomeasure s on G whose singular support is E; and that if no portion of E is a Helson set, then such an s may be chosen having its support equal to E. There follow (Corollaries (2.2.4) and (2.3.2)) sufficient conditions for the relations to hold for some pseudomeasure s, E and F being given closed subsets of G. These results are analogues and refinements of a theorem of Pollard [4] for the case G = R, which asserts the existence of a function in L∞(R) whose spectrum coincides with any preassigned closed subset of R.

The uniform compactification of a locally compact abelian group

1990 ◽  
Vol 108 (3) ◽  
pp. 527-538 ◽  
Author(s):  
M. Filali
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Complex Functions ◽  
Discrete Semigroups ◽  
Continuous Complex ◽  
Image Position ◽  
The Given ◽  
Uniformly Continuous

In recent years, the Stone-Čech compactification of certain semigroups (e.g. discrete semigroups) has been an interesting semigroup compactification (i.e. a compact right semitopological semigroup which contains a dense continuous homomorphic image of the given semigroup) to study, because an Arens-type product can be introduced. If G is a non-compact and non-discrete locally compact abelian group, then it is not possible to introduce such a product into the Stone-Čech compactification βG of G (see [1]). However, let UC(G) be the Banach algebra of bounded uniformly continuous complex functions on G, and let UG be the spectrum of UC(G) with the Gelfand topology. If f∈ UC(G), then the functions f and fy defined on G byare also in UC(G).

scholarly journals Local Complements to the Hausdorff-Young Theorem for Amalgams

1987 ◽  
Vol 30 (3) ◽  
pp. 325-333
Author(s):  
Maria L. Torres de Squire
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact

AbstractLet G be a locally compact abelian group. An amalgam space (Lp ℓq)(G) (1 ≦ p,q ≦ ∞) is a Banach space of functions which belong locally to LP(G) and globally to ℓq. In this paper we present noninclusion results related to the Hausdorff-Young theorem for amalgams.

A diophantine problem on groups. III

1971 ◽  
Vol 70 (1) ◽  
pp. 31-47 ◽  
Author(s):  
R. C. Baker
Keyword(s):  
Abelian Group ◽  
Finite Order ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Diophantine Problem ◽  
Precise Sense ◽  
Image Position ◽  
Almost All

AbstractThe following generalization of a theorem of Weyl appeared in part I of this series of papers. Let G be a locally compact Abelian group with dual group ĝ. Let be a sequence in ĝ, not too slowly growing in a certain precise sense. Then, provided ĝ has ‘not too many’ elements of finite order, the sequencesare uniformly distributed on the circle, for almost all x in G.

scholarly journals Lifting mixing properties by Rokhlin cocycles

2011 ◽  
Vol 32 (2) ◽  
pp. 763-784 ◽  
Author(s):  
MARIUSZ LEMAŃCZYK ◽  
FRANÇOIS PARREAU
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Product Space ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Mixing Properties ◽  
Skew Products ◽  
Image Position

AbstractWe study the problem of lifting various mixing properties from a base automorphismT∈Aut(X,ℬ,μ) to skew products of the formTφ,𝒮, where φ:X→Gis a cocycle with values in a locally compact Abelian groupG, 𝒮=(Sg)g∈Gis a measurable representation ofGinAut(Y,𝒞,ν) andTφ,𝒮acts on the product space (X×Y,ℬ⊗𝒞,μ⊗ν) byIt is also shown that wheneverTis ergodic (mildly mixing, mixing) butTφ,𝒮is not ergodic (is not mildly mixing, not mixing), then, on a non-trivial factor 𝒜⊂𝒞 of 𝒮, the corresponding Rokhlin cocyclex↦Sφ(x)∣𝒜is a coboundary (a quasi-coboundary).

Uniform approximation by Fourier–Stieltjes transforms

1968 ◽  
Vol 64 (2) ◽  
pp. 323-333 ◽  
Author(s):  
Donald E. Ramirez
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Continuous Functions ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Norm Topology ◽  
Locally Compact ◽  
Borel Measures ◽  
Bounded Functions ◽  
Stieltjes Transforms

Let G be a locally compact Abelian group; Γ the dual group of G; CB(Γ) the algebra of continuous, bounded functions on Γ C0(Γ) the algebra of continuous functions on Γ which vanish at infinity; M(G) the algebra of bounded Borel measures on G; M(G)^ the algebra of Fourier–Stieltjes transforms; and M(G)^− the completion of M(G)^ in the sup-norm topology on Γ. The object of this paper is to study the natural pairing between M(G)^ and M(Γ).

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