The inductive limit of uncountably many compact Abelian groups

1968 ◽  
Vol 64 (4) ◽  
pp. 985-987
Author(s):  
D. L. Salinger
Keyword(s):  
Inductive Limit ◽  
Abelian Groups ◽  
Topological Groups ◽  
Inductive Limits ◽  
Locally Compact ◽  
Linear Spaces ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Topological Linear

In (2), Varopoulos examined the structure of topological groups that are the inductive limits of countably many locally compact Abelian groups. The purpose of this note is to show that the theory does not extend to the case of uncountably many groups. We give two examples, the first to show that the strict inductive limit of uncountably many compact Abelian groups need not be complete, the second to show it need not be separated by its continuous characters. The treatment of this latter half follows closely that given for topological linear spaces by Douady in (l).

A theorem on cardinal numbers associated with inductive limits of locally compact Abelian groups

1965 ◽  
Vol 61 (1) ◽  
pp. 69-74 ◽  
Author(s):  
J. B. Reade
Keyword(s):  
Classical Theory ◽  
Inductive Limit ◽  
Abelian Groups ◽  
Topological Groups ◽  
Inductive Limits ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Cardinal Numbers ◽  
Compact Abelian Groups

Our motivation for this paper is to be found in (2) and (3). In (2) Varopoulos considers inductive limits of topological groups, in particular what he calls ‘ℒ∞’. (He calls a topology an ℒ∞-topology when it is the inductive limit of a decreasing sequence of locally compact Hausdorff topologies.) In (2) he proves that much of the classical theory of locally compact Abelian groups also goes through for Abelian ℒ∞-groups, in particular Pontrjagin duality.

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 Isomorphisms of some convolution algebras and their multiplier algebras

1972 ◽  
Vol 7 (3) ◽  
pp. 321-335 ◽  
Author(s):  
U.B. Tewari ◽  
G.I. Gaudry
Keyword(s):  
Abelian Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Algebra Isomorphism ◽  
Locally Compact Abelian Groups ◽  
Convolution Algebras ◽  
Compact Abelian Groups ◽  
Multiplier Algebras

Let G1 and G2 be two locally compact abelian groups and let 1 ≤ p ∞. We prove that G1 and G2 are isomorphic as topological groups provid∈d there exists a bipositive or isometric algebra isomorphism of M(Ap (G1)) onto M(Ap (G2)). As a consequence of this, we prove that G1 and G2 are isomorphic as topological groups provided there exists a bipositive or isometric algebra isomorphism of Ap (G1) onto Ap (G2). Similar results about the algebras L1 ∩ Lp and L1 ∩ C0 are also established.

Scaling sets and generalized scaling sets on Cantor dyadic group

2020 ◽  
Vol 18 (04) ◽  
pp. 2050019
Author(s):  
Prasadini Mahapatra ◽  
Divya Singh
Keyword(s):  
Abelian Groups ◽  
Real Case ◽  
Locally Compact ◽  
Wavelet Sets ◽  
Locally Compact Abelian Groups ◽  
Dyadic Group ◽  
Compact Abelian Groups ◽  
Cantor Dyadic Group

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group [Formula: see text]. These results can further be generalized to arbitrary locally compact abelian groups.

Singular measures with absolutely continuous convolution squares

1966 ◽  
Vol 62 (3) ◽  
pp. 399-420 ◽  
Author(s):  
Edwin Hewitt ◽  
Herbert S. Zuckerman
Keyword(s):  
Abelian Groups ◽  
Natural Order ◽  
Continuous Measure ◽  
Locally Compact ◽  
Absolutely Continuous ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Singular Measures ◽  
Compact Abelian Groups ◽  
Nikodym Derivative

Introduction. A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivašev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such thatfor every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has pth power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.

Sign in / Sign up

Export Citation Format

Share Document