A CONVOLUTION-INDUCED TOPOLOGY ON THE ORLICZ SPACE OF A LOCALLY COMPACT GROUP

2015 ◽  
Vol 99 (1) ◽  
pp. 1-11
Author(s):  
IBRAHIM AKBARBAGLU ◽  
SAEID MAGHSOUDI
Keyword(s):  
Compact Group ◽  
Orlicz Space ◽  
Haar Measure ◽  
Locally Compact Group ◽  
Young Function ◽  
Strict Topology ◽  
Locally Compact ◽  
Induced Topology ◽  
Locally Convex Topology ◽  
Locally Convex

Let $G$ be a locally compact group with a fixed left Haar measure. In this paper, given a strictly positive Young function ${\rm\Phi}$, we consider $L^{{\rm\Phi}}(G)$ as a Banach left $L^{1}(G)$-module. Then we equip $L^{{\rm\Phi}}(G)$ with the strict topology induced by $L^{1}(G)$ in the sense of Sentilles and Taylor. Some properties of this locally convex topology and a comparison with weak$^{\ast }$, bounded weak$^{\ast }$ and norm topologies are presented.

Translation-Invariant Operators On Lp(G), 0 < p < 1 (II)

1977 ◽  
Vol 29 (3) ◽  
pp. 626-630 ◽  
Author(s):  
Daniel M. Oberlin
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Lebesgue Space ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Translation Invariant ◽  
Translation Operators ◽  
Left And Right

For a locally compact group G, let LP(G) be the usual Lebesgue space with respect to left Haar measure m on G. For x ϵ G define the left and right translation operators Lx and Rx by Lx f(y) = f(xy), Rx f(y) = f(yx)(f ϵ Lp(G),y ϵ G). The purpose of this paper is to prove the following theorem.

Counterexample to a Conjecture of Greenleaf

1970 ◽  
Vol 13 (4) ◽  
pp. 497-499 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Image Position

Greenleaf states the following conjecture in [1, p. 69]. Let G be a (connected, separable) amenable locally compact group with left Haar measure, μ, and let U be a compact symmetric neighbourhood of the unit. Then the sets, {Um}, have the following property: given ɛ > 0 and compact K ⊂ G, ∃ m0 = m0(ɛ, K) such that

On certain porous sets in the Orlicz space of a locally compact group

2012 ◽  
Vol 129 (1) ◽  
pp. 99-111 ◽  
Author(s):  
Ibrahim Akbarbaglu ◽  
Saeid Maghsoudi
Keyword(s):  
Compact Group ◽  
Orlicz Space ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Porous Sets

scholarly journals A Note on the Kawada-into Theorem

1963 ◽  
Vol 13 (4) ◽  
pp. 295-296 ◽  
Author(s):  
John S. Pym
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Almost Everywhere ◽  
Regular Borel Measure ◽  
Image Position ◽  
Complex Valued

If µ is a bounded regular Borel measure on a locally compact group G, and L1(G) denotes the class of complex-valued functions which are integrable with respect to the left Haar measure m of G, then, for each f∈L1(G),defines almost everywhere (a.e.) with respect to m a function μ*f which is again in L1(G). The measure μ will be called isotone on G mapping f→μ*f is isotone, i.e. f≧0 a.e. (m) if and only if μ*f≧0 a.e. (m).

HYPERGROUP ALGEBRAS AS TOPOLOGICAL ALGEBRAS

2014 ◽  
Vol 90 (3) ◽  
pp. 486-493
Author(s):  
S. MAGHSOUDI ◽  
J. B. SEOANE-SEPÚLVEDA
Keyword(s):  
Haar Measure ◽  
Lebesgue Space ◽  
Topological Algebra ◽  
Topological Algebras ◽  
Locally Compact ◽  
Continuous Functionals ◽  
Locally Convex Topology ◽  
Compact Hypergroup ◽  
Locally Convex ◽  
Hypergroup Algebras

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$ be a locally compact hypergroup endowed with a left Haar measure and let $L^1(K)$ be the usual Lebesgue space of $K$ with respect to the left Haar measure. We investigate some properties of $L^1(K)$ under a locally convex topology $\beta ^1$. Among other things, the semireflexivity of $(L^1(K), \beta ^1)$ and of sequentially$\beta ^1$-continuous functionals is studied. We also show that $(L^1(K), \beta ^1)$ with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if $K$ is compact.

scholarly journals A Duality of Locally Compact Groups That Does Not Involve the Haar Measure

2015 ◽  
Vol 116 (2) ◽  
pp. 250 ◽  
Author(s):  
Yulia Kuznetsova
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
C Algebras ◽  
Commutative Algebras

We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf $C^*$-algebras, and a similar map on the category of coinvolutive Hopf-von Neumann algebras. In the $C^*$-version, this functor sends $C_0(G)$ to $C^*(G)$ and vice versa, for every locally compact group $G$. As opposed to preceding approaches, there is an explicit description of commutative and co-commutative algebras in the range of this map (without assumption of being isomorphic to their bidual): these algebras have the form $C_0(G)$ or $C^*(G)$ respectively, where $G$ is a locally compact group. The von Neumann version of the functor puts into duality, in the group case, the enveloping von Neumann algebras of the algebras above: $C_0(G)^{**}$ and $C^*(G)^{**}$.

Convolutions as Bilinear and Linear Operators

1964 ◽  
Vol 16 ◽  
pp. 275-285 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Linear Operators ◽  
Locally Compact ◽  
Radon Measures

Throughout this paper X denotes a fixed Hausdorff locally compact group with left Haar measure dx. Various spaces of functions and measures on X will recur in the discussion, so we name and describe them forthwith. All functions and measures on X will be scalarvalued, though it matters little whether the scalars are real or complex.C = C(X) is the space of all continuous functions on X, Cc = Cc(X) its subspace formed of functions with compact supports. M = M(X) denotes the space of all (Radon) measures on X, Mc = MC(X) the subspace formed of those measures with compact supports. In general we denote the support of a function or a measure ξ by [ξ].

Locally convex algebras which determine a locally compact group

2016 ◽  
pp. 1-11
Author(s):  
Gholam Hossein Esslamzadeh ◽  
Hossein Javanshiri ◽  
Rasoul Nasr-Isfahani
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Locally Convex Algebras ◽  
Locally Convex

scholarly journals A CAT(0)-VALUED POINTWISE ERGODIC THEOREM

2011 ◽  
Vol 03 (02) ◽  
pp. 145-152 ◽  
Author(s):  
TIM AUSTIN
Keyword(s):  
Compact Group ◽  
Ergodic Theorem ◽  
Haar Measure ◽  
Probability Space ◽  
Invariant Function ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Pointwise Ergodic Theorem ◽  
Measurable Map ◽  
Compact Subsets

This note proves a version of the pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space. The precise setting consists of an amenable locally compact group G with left Haar measure mG, a jointly measurable, probability-preserving action [Formula: see text] of G on a probability space, and a separable complete CAT(0)-space (X, d) with barycentre map b. In this setting we show that if (Fn)n ≥ 1 is a tempered Følner sequence of compact subsets of G and f : Ω → X is a measurable map such that for some (and hence any) fixed x ∈ X, we have [Formula: see text] then as n → ∞ the functions of empirical barycentres [Formula: see text] converge pointwise for almost every ω to a T-invariant function [Formula: see text].

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