scholarly journals The convolution-induced topology onL∞(G)and linearly dependent translates inL1(G)

1982 ◽  
Vol 5 (1) ◽  
pp. 11-20 ◽  
Author(s):  
G. Crombez ◽  
W. Govaerts
Keyword(s):  
Weak Topology ◽  
Trigonometric Polynomials ◽  
Major Result ◽  
Almost Periodic ◽  
Convolution Operators ◽  
Norm Topology ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Hausdorff Group ◽  
Induced Topology

Given a locally compact Hausdorff groupG, we consider onL∞(G)theτc-topology, i.e. the weak topology under all convolution operators induced by functions inL1(G). As a major result we characterize the trigonometric polynomials on a compact group as those functions inL1(G)whose left translates are contained in a finite-dimensional set. From this, we deduce thatτcis different from thew∗-topology onL∞(G)wheneverGis infinite. As another result, we show thatτccoincides with the norm-topology if and only ifGis discrete. The properties ofτcare then studied further and we pay attention to theτc-almost periodic elements ofL∞(G).

scholarly journals Finite dimensional H-invariant spaces

1997 ◽  
Vol 56 (3) ◽  
pp. 353-361
Author(s):  
K.E. Hare ◽  
J.A. Ward
Keyword(s):  
Compact Group ◽  
Closed Subspace ◽  
Sufficient Conditions ◽  
Trigonometric Polynomials ◽  
Left Translation ◽  
Almost Periodic ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Invariant Spaces

A subset V of M(G) is left H-invariant if it is invariant under left translation by the elements of H, a subset of a locally compact group G. We establish necessary and sufficient conditions on H which ensure that finite dimensional subspaces of M(G) when G is compact, or of L∞(G) when G is locally compact Abelian, which are invariant in this weaker sense, contain only trigonometric polynomials. This generalises known results for finite dimensional G-invariant subspaces. We show that if H is a subgroup of finite index in a compact group G, and the span of the H-translates of μ is a weak*-closed subspace of L∞(G) or M(G) (or is closed in Lp(G)for 1 ≤ p < ∞), then μ is a trigonometric polynomial.We also obtain some results concerning functions that possess the analogous weaker almost periodic condition relative to H.

The τc -topology on locally compact foundation semigroups

2009 ◽  
Vol 46 (1) ◽  
pp. 25-35
Author(s):  
Ali Ghaffari
Keyword(s):  
Topological Semigroup ◽  
Weak Topology ◽  
The Other ◽  
Bounded Subset ◽  
Almost Periodic ◽  
Norm Topology ◽  
Locally Compact

Given a foundation locally compact Hausdorff topological semigroup S , we consider on Ma ( S )* the τc -topology, i.e. the weak topology under all right multipliers induced by measures in Ma ( S ). For such an arbitrary S the τc -topology is not weaker than the weak*-topology and not stronger than the norm topology on Ma ( S )*. However, a further investigation shows that for compact S the norm topology and τc -topology coincide on every norm bounded subset of Ma ( S ). Among the other results we mention that except for discrete S the τc -topology is always different from the norm-topology. Finally, we give some results about τc -almost periodic functionals.

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

scholarly journals Density and representation theorems for multipliers of type (p, q)

1967 ◽  
Vol 7 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Alessandro Figà-Talamanca ◽  
G. I. Gaudry
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Important Class ◽  
Lebesgue Spaces ◽  
Representation Theorems ◽  
Locally Compact ◽  
Bounded Linear ◽  
Character Group ◽  
Hausdorff Group ◽  
Lca Group

Let G be a locally compact Abelian Hausdorff group (abbreviated LCA group); let X be its character group and dx, dx be the elements of the normalised Haar measures on G and X respectively. If 1 < p, q < ∞, and Lp(G) and Lq(G) are the usual Lebesgue spaces, of index p and q respectively, with respect to dx, a multiplier of type (p, q) is defined as a bounded linear operator T from Lp(G) to Lq(G) which commutes with translations, i.e. τxT = Tτx for all x ∈ G, where τxf(y) = f(x+y). The space of multipliers of type (p, q) will be denoted by Lqp. Already, much attention has been devoted to this important class of operators (see, for example, [3], [4], [7]).

scholarly journals Uniqueness theorem for Fourier transformable measures on LCA groups

2020 ◽  
Vol 54 (2) ◽  
pp. 211-219
Author(s):  
S.Yu. Favorov
Keyword(s):  
Uniqueness Theorem ◽  
Real Axis ◽  
Almost Periodic ◽  
Periodic Measure ◽  
Locally Compact ◽  
The Real ◽  
Approach Infinity ◽  
The Masses ◽  
Lca Groups

We show that if points of supports of two discrete ”not very thick” Fourier transformable measures on locally compact abelian (LCA) groups tend to one another at infinity and the same is true for the masses at these points, then these measures coincide. The result is valid for discrete almost periodic measures on LCA groups too. Also, we show that the result is false for some discrete ”thick” measures. To do this, we construct a discrete almost periodic measure on the real axis, whose masses at the points of support tend to zero as these points approach infinity.

scholarly journals Survey on the Figa`–Talamanca Herz algebra †

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 660
Author(s):  
Antoine Derighetti
Keyword(s):  
Finite Groups ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Convolution Operators ◽  
Locally Compact

This paper presents a self contained approach to the theory of convolution operators on locally compact groups (both commutative and non commutative) based on the use of the Fig a ` –Talamanca Herz algebras. The case of finite groups is also considered.

Continuous Ergodic Extensions and Fibre Bundles

1978 ◽  
Vol 30 (02) ◽  
pp. 373-391 ◽  
Author(s):  
Robert J. Zimmer
Keyword(s):  
Real Line ◽  
Lebesgue Space ◽  
Transformation Group ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Fibre Bundles ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Measure Preserving Transformation

If a locally compact group G acts as a measure preserving transformation group on a Lebesgue space X, then there is a naturally induced unitary representation of G on L2(X), and one can study the action on X by means of this representation. The situation in which the representation has discrete spectrum (i.e., is the direct sum of finite dimensional representations) and the action is ergodic was examined by von Neumann and Halmos when G is the integers or the real line [7], and by Mackey for general non-abelian G [10].

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