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.

scholarly journals Arc components in locally compact groups are Borel sets

2002 ◽  
Vol 65 (1) ◽  
pp. 1-8
Author(s):  
Karl Heinrich Hofmann
Keyword(s):  
Compact Group ◽  
Set Theory ◽  
Abelian Groups ◽  
Affirmative Answer ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Borel Sets ◽  
Models Of Set Theory

Are the arc components in a locally compact group Borel subsets? An affirmative answer is provide for locally compact groups satisfying the First Axiom of Count-ability. For general locally compact groups the question is reduced to compact connected Abelian groups. In certain models of set theory the answer is negative.

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.

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 THE MODULUS OF WEAKLY COMPACT MULTIPLIERS ON THE BANACH ALGEBRA L1(𝒢)** OF A LOCALLY COMPACT GROUP

2012 ◽  
Vol 86 (2) ◽  
pp. 315-321
Author(s):  
MOHAMMAD JAVAD MEHDIPOUR
Keyword(s):  
Banach Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Locally Compact ◽  
Weakly Compact ◽  
Necessary And Sufficient ◽  
Funct Anal

AbstractIn this paper we give a necessary and sufficient condition under which the answer to the open problem raised by Ghahramani and Lau (‘Multipliers and modulus on Banach algebras related to locally compact groups’, J. Funct. Anal. 150 (1997), 478–497) is positive.

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 Ends of locally compact groups and their coset spaces

1974 ◽  
Vol 17 (3) ◽  
pp. 274-284 ◽  
Author(s):  
C. H. Houghton
Keyword(s):  
Compact Group ◽  
Direct Product ◽  
Compact Space ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Countable Base ◽  
Locally Compact ◽  
Coset Spaces ◽  
Compact Boundary

Freudenthal [5, 7] defined a compactification of a rim-compact space, that is, a space having a base of open sets with compact boundary. The additional points are called ends and Freudenthal showed that a connected locally compact non-compact group having a countable base has one or two ends. Later, Freudenthal [8], Zippin [16], and Iwasawa [11] showed that a connected locally compact group has two ends if and only if it is the direct product of a compact group and the reals.

scholarly journals A note on the duality of locally compact groups

1968 ◽  
Vol 9 (2) ◽  
pp. 87-91 ◽  
Author(s):  
J. W. Baker
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Group Topology ◽  
Locally Compact ◽  
Natural Way

Let H be a group of characters on an (algebraic) abelian group G. In a natural way, we may regard G as a group of characters on H. In this way, we obtain a duality between the two groups G and H. One may pose several problems about this duality. Firstly, one may ask whether there exists a group topology on G for which H is precisely the set of continuous characters. This question has been answered in the affirmative in [1]. We shall say that such a topology is compatible with the duality between G and H. Next, one may ask whether there exists a locally compact group topology on G which is compatible with a given duality and, if so, whether there is more than one such topology. It is this second question (previously considered by other authors, to whom we shall refer below) which we shall consider here.

Exponential Boundedness and Amenability of Open Subsemigroups of Locally Compact Groups

1994 ◽  
Vol 46 (06) ◽  
pp. 1263-1274 ◽  
Author(s):  
Wojciech Jaworski
Keyword(s):  
Probability Measure ◽  
Open Subset ◽  
Compact Support ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Complete Proof ◽  
Absolutely Continuous ◽  
Crucial Part

Abstract Let G be a connected amenable locally compact group with left Haar measure λ. In an earlier work Jenkins claimed that exponential boundedness of G is equivalent to each of the following conditions: (a) every open subsemigroup S ⊆ G is amenable; (b) given and a compact K ⊆ G with nonempty interior there exists an integer n such that (c) given a signed measure of compact support and nonnegative nonzero f ∈ L ∞(G), the condition v * f ≥ 0 implies v(G) ≥ 0. However, Jenkins‚ proof of this equivalence is not complete. We give a complete proof. The crucial part of the argument relies on the following two results: (1) an open σ-compact subsemigroup S ⊆ G is amenable if and only if there exists an absolutely continuous probability measure μ on S such that lim for every s ∈ S; (2) G is exponentially bounded if and only if for every nonempty open subset U ⊆ G.

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

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