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 Hereditarily minimal topological groups

2019 ◽  
Vol 31 (3) ◽  
pp. 619-646 ◽  
Author(s):  
Wenfei Xi ◽  
Dikran Dikranjan ◽  
Menachem Shlossberg ◽  
Daniele Toller
Keyword(s):  
Abelian Groups ◽  
Multiplicative Group ◽  
The Other ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Abelian Case ◽  
Compact Abelian Groups ◽  
Minimal Groups

Abstract We study locally compact groups having all subgroups minimal. We call such groups hereditarily minimal. In 1972 Prodanov proved that the infinite hereditarily minimal compact abelian groups are precisely the groups {\mathbb{Z}_{p}} of p-adic integers. We extend Prodanov’s theorem to the non-abelian case at several levels. For infinite hypercentral (in particular, nilpotent) locally compact groups, we show that the hereditarily minimal ones remain the same as in the abelian case. On the other hand, we classify completely the locally compact solvable hereditarily minimal groups, showing that, in particular, they are always compact and metabelian. The proofs involve the (hereditarily) locally minimal groups, introduced similarly. In particular, we prove a conjecture by He, Xiao and the first two authors, showing that the group {\mathbb{Q}_{p}\rtimes\mathbb{Q}_{p}^{*}} is hereditarily locally minimal, where {\mathbb{Q}_{p}^{*}} is the multiplicative group of non-zero p-adic numbers acting on the first component by multiplication. Furthermore, it turns out that the locally compact solvable hereditarily minimal groups are closely related to this group.

Embeddings of locally compact abelian p-groups in Hawaiian groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yanga Bavuma ◽  
Francesco G. Russo
Keyword(s):  
Algebraic Topology ◽  
Geometric Interpretation ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Hawaiian Earring ◽  
Path Connected

Abstract We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

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 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.

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).

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