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 invariant convex subsets in algebras defined on a locally compact group G

2012 ◽  
Vol 49 (3) ◽  
pp. 301-314
Author(s):  
Ali Ghaffari
Keyword(s):  
Compact Group ◽  
Sufficient Conditions ◽  
Locally Compact Group ◽  
Left Translation ◽  
Measure Algebra ◽  
Locally Compact ◽  
Translation Invariant ◽  
Second Dual ◽  
Necessary And Sufficient ◽  
Convex Subsets

Suppose that A is either the Banach algebra L1(G) of a locally compact group G, or measure algebra M(G), or other algebras (usually larger than L1(G) and M(G)) such as the second dual, L1(G)**, of L1(G) with an Arens product, or LUC(G)* with an Arenstype product. The left translation invariant closed convex subsets of A are studied. Finally, we obtain necessary and sufficient conditions for LUC(G)* to have 1-dimensional left ideals.

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).

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].

Groups with Equal Uniformities

1969 ◽  
Vol 21 ◽  
pp. 655-659 ◽  
Author(s):  
R. T. Ramsay
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Vector Group ◽  
Locally Compact ◽  
Left And Right ◽  
The Relationship

If G = (G, τ) is a topological group with topology τ, then there is a smallest topology τ* ⊇ τ such that G* = (G, τ*) is a topological group with equal left and right uniformities (1). Bagley and Wu introduced this topology in (1), and studied the relationship between Gand G*. In this paper we prove some additional results concerning G* and groups with equal uniformities in general. The structure of locally compact groups with equal uniformities has been studied extensively. If G is a locally compact connected group, then G has equal uniformities if and only if G ≅ V× K,where F is a vector group and Kis a compact group (5). More generally, every locally compact group with equal uniformities has an open normal subgroup of the form V× K(4).

Square-integrable representations and multipliers

2019 ◽  
Vol 26 (4) ◽  
pp. 505-513
Author(s):  
Gerhard Racher
Keyword(s):  
Compact Group ◽  
Von Neumann Algebra ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Translation Invariant ◽  
Von Neumann

Abstract We observe a connection between the existence of square-integrable representations of a locally compact group G and the existence of nonzero translation invariant operators from its Fourier–Stieltjes algebra {B(G)} into {L^{2}(G)} or, equivalently, from {L^{2}(G)} into its enveloping von Neumann algebra {C^{*}(G)^{**}} .

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].

Bipositive and Isometric Isomorphisms of Some Convolution Algebras

1965 ◽  
Vol 17 ◽  
pp. 839-846 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Compact Group ◽  
Lebesgue Space ◽  
Inductive Limit ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Natural Topology ◽  
Locally Compact ◽  
Convolution Algebras ◽  
Image Position ◽  
Complex Valued

Throughout this paper the term "space" will mean "Hausdorff locally compact space" and the term '"group" will mean "Hausdorff locally compact group." If G is a group and 1 ≤ p < ∞, Lp(G) denotes the usual Lebesgue space formed relative to left Haar measure on G. It is well known that L1(G) is an algebra under convolution, and that the same is true of Lp(G) whenever G is compact. We introduce also the space Cc(G) of complex-valued continuous functions f on G for each of which the support (supp f), is compact. The "natural" topology of CC(G) is obtained by regarding CC(G) as the inductive limit of its subspaces

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