Equidimensional immersions of locally compact groups

1989 ◽  
Vol 105 (2) ◽  
pp. 253-261 ◽  
Author(s):  
K. H. Hofmann ◽  
T. S. Wu ◽  
J. S. Yang
Keyword(s):  
Exponential Function ◽  
Lie Group ◽  
Compact Groups ◽  
Locally Compact ◽  
Lie Group Theory ◽  
Group A ◽  
Image Position ◽  
Analytic Subgroup ◽  
Algebra Level ◽  
Connected Lie Group

Dense immersions occur frequently in Lie group theory. Suppose that exp: g → G denotes the exponential function of a Lie group and a is a Lie subalgebra of g. Then there is a unique Lie group ALie with exponential function exp:a → ALie and an immersion f:ALie→G whose induced morphism L(j) on the Lie algebra level is the inclusion a → g and which has as image an analytic subgroup A of G. The group Ā is a connected Lie group in which A is normal and dense and the corestrictionis a dense immersion. Unless A is closed, in which case f' is an isomorphism of Lie groups, dim a = dim ALie is strictly smaller than dim h = dim H.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals Topological entropy in totally disconnected locally compact groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2163-2186 ◽  
Author(s):  
ANNA GIORDANO BRUNO ◽  
SIMONE VIRILI
Keyword(s):  
Topological Entropy ◽  
Sufficient Conditions ◽  
Invariant Subgroup ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Totally Disconnected ◽  
Dynamical Interpretation

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$. We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$. As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$. Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.

scholarly journals Rudin-Shapiro sequences for arbitrary compact groups

1976 ◽  
Vol 22 (4) ◽  
pp. 421-430 ◽  
Author(s):  
J. R. McMullen ◽  
J. F. Price
Keyword(s):  
Compact Group ◽  
Trigonometric Polynomials ◽  
Compact Groups ◽  
Group A ◽  
Image Position

AbstractLet G be a compact group. A sequence of functions in L∞ (G) is said to be a Rudin-Shapiro sequence (briefly, an RS-sequence) if the following conditions hold: (1) (2) (3) The main purpose here is to prove the following theorem: Theorem: Theorem. Let G be an infinite compact group. Then G has an RS-sequence consisting of trigonometric polynomials.

Bernstein's theorem for compact, connected Lie groups

1986 ◽  
Vol 99 (2) ◽  
pp. 297-305 ◽  
Author(s):  
Garth I. Gaudry ◽  
Rita Pini
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Image Position ◽  
Complex Valued ◽  
Connected Lie Group

In what follows, G will denote a compact, connected Lie group.Definition. A complex-valued function f ε L2(G) lies in the space A(G) if it can be written in the formwhereThe A(G) norm of f is the infimum of all sums (2) with respect to all decompositions (1).

Compactness of a Locally Compact Group G and Geometric Properties of Ap(G)

1996 ◽  
Vol 48 (6) ◽  
pp. 1273-1285 ◽  
Author(s):  
Tianxuan Miao
Keyword(s):  
Topological Group ◽  
Compact Group ◽  
Multiplication Operator ◽  
Locally Compact Group ◽  
Compact Groups ◽  
Locally Compact ◽  
Geometric Properties ◽  
Nikodym Property ◽  
Group A ◽  
Locally Compact Topological Group

AbstractLet G be a locally compact topological group. A number of characterizations are given of the class of compact groups in terms of the geometric properties such as Radon-Nikodym property, Dunford-Pettis property and Schur property of Ap(G), and the properties of the multiplication operator on PFp(G). We extend and improve several results of Lau and Ülger [17] to Ap(G) and Bp(G) for arbitrary p.

Locally compact groups: maximal compact subgroups and N-groups

1988 ◽  
Vol 104 (1) ◽  
pp. 47-64
Author(s):  
R. W. Bagley ◽  
T. S. Wu ◽  
J. S. Yang
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Compact Set ◽  
Locally Compact ◽  
Group A ◽  
Totally Disconnected ◽  
Compactly Generated

AbstractIf G is a locally compact group such thatG/G0contains a uniform compactly generated nilpotent subgroup, thenGhas a maximal compact normal subgroupKsuch thatG/Gis a Lie group. A topological groupGis an N-group if, for each neighbourhoodUof the identity and each compact setC⊂G, there is a neighbourhoodVof the identity such thatfor eachg∈G. Several results on N-groups are obtained and it is shown that a related weaker condition is equivalent to local finiteness for certain totally disconnected groups.

On nth roots and infinitely divisible elements in a connected Lie group

1981 ◽  
Vol 89 (2) ◽  
pp. 293-299 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Finite Number ◽  
Lie Groups ◽  
Lie Group ◽  
Connected Components ◽  
Compact Lie Group ◽  
Infinitely Divisible ◽  
Closed Set ◽  
Image Position ◽  
Connected Lie Group

For any group G, x ∈ G and n ∈ ℕ (the natural numbers), leti.e. the set of all nth roots of x in G. If G is a Hausdorff topological group, then Rn(x, G) is a closed set in G, but may otherwise be quite complicated. However, as we have observed in (4), if G is a compact Lie group, then Rn(x, G) always has a finite number of connected components, and this result has led us to wonder about the connectedness properties of Rn(x, G) for other Lie groups G. Here is the result.

scholarly journals Splitting definably compact groups in o-minimal structures

2011 ◽  
Vol 76 (3) ◽  
pp. 973-986
Author(s):  
Marcello Mamino
Keyword(s):  
Lie Group ◽  
Cartesian Product ◽  
Real Closed Field ◽  
Compact Groups ◽  
Closed Field ◽  
Parallel Result ◽  
Derived Subgroup ◽  
Connected Lie Group

AbstractAn argument of A. Borel [Bor-61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

Locally compact groups whose conjugation representations satisfy a Kazhdan type property or have countable support

1994 ◽  
Vol 116 (1) ◽  
pp. 79-97 ◽  
Author(s):  
Eberhard Kaniuth ◽  
Annette Markfort
Keyword(s):  
Modular Function ◽  
Equivalence Classes ◽  
Irreducible Representations ◽  
Compact Groups ◽  
Locally Compact ◽  
Countable Support ◽  
Type Property ◽  
Invariant Means ◽  
Image Position ◽  
Conjugation Representation

For a locally compact group G with left Haar measure and modular function δ the conjugation representation γG of G on L2(G) is defined byf ∈ L2(G), x, y ∈ G. γG has been investigated recently (see [19, 20, 21, 24, 32, 35]). For semi-simple Lie groups, a related representation has been studied in [25]. γG is of interest not least because of its connection to questions on inner invariant means on L∞(G). In what follows suppγG denotes the support of γG in the dual space Ĝ, that is the closed subset of all equivalence classes of irreducible representations which are weakly contained in γG. The purpose of this paper is to establish relations between properties such as a variant of Kazhdan's property and discreteness or countability of supp γG and the structure of G.

Central Idempotent Measures on Unitary Groups

1970 ◽  
Vol 22 (4) ◽  
pp. 719-725 ◽  
Author(s):  
Daniel Rider
Keyword(s):  
Bounded Function ◽  
Abelian Groups ◽  
Locally Compact Group ◽  
Unitary Groups ◽  
Compact Groups ◽  
Central Idempotent ◽  
Continuous Bounded Function ◽  
Locally Compact ◽  
Borel Measures ◽  
Image Position

Let G be a locally compact group and M(G) the space of finite regular Borel measures on G. If μ and v are in M(G), their convolution is defined byThus, if f is a continuous bounded function on G,μ is central if μ(Ex) = μ(xE) for all x ∈ G and all measurable sets E. μ is idempotent if μ * μ = μ.The idempotent measures for abelian groups have been classified by Cohen [1]. In this paper we will show that for a certain class of compact groups, containing the unitary groups, the central idempotents can be characterized. The method consists of showing that, in these cases, the central idempotents arise from idempotents on abelian groups and applying Cohen's result.

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