scholarly journals Measures with unique tangent measures in metric groups

2005 ◽  
Vol 97 (2) ◽  
pp. 298 ◽  
Author(s):  
Pertti Mattila
Keyword(s):  
Haar Measure ◽  
Radon Measure ◽  
Invariant Subgroup ◽  
Locally Compact ◽  
Almost Everywhere ◽  
Tangent Measure ◽  
Metric Group ◽  
Metric Groups

We show that a Radon measure on a locally compact metric group with natural dilations has almost everywhere a unique tangent measure if and only if it has almost everywhere a Haar measure of a closed dilation invariant subgroup as its unique tangent measure.

scholarly journals Characterization of Relative Domination Principle

1973 ◽  
Vol 50 ◽  
pp. 175-184 ◽  
Author(s):  
Isao Higuchi ◽  
Masayuki Itô
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Radon Measure ◽  
Convolution Type ◽  
Convolution Kernel ◽  
Locally Compact ◽  
Positive Radon Measure ◽  
Convolution Kernels

Let X be a locally compact and σ-compact Abelian group and ξ be the Haar measure of X. A positive Radon measure N on X is called a convolution kernel when we regard it as a kernel of potentials of convolution type. M. Itô [4], [6] characterized the convolution kernel which satisfies the domination principle. The purpose of this paper is to characterize the relative domination principle for the convolution kernels.

scholarly journals A Note on the Kawada-into Theorem

1963 ◽  
Vol 13 (4) ◽  
pp. 295-296 ◽  
Author(s):  
John S. Pym
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Almost Everywhere ◽  
Regular Borel Measure ◽  
Image Position ◽  
Complex Valued

If µ is a bounded regular Borel measure on a locally compact group G, and L1(G) denotes the class of complex-valued functions which are integrable with respect to the left Haar measure m of G, then, for each f∈L1(G),defines almost everywhere (a.e.) with respect to m a function μ*f which is again in L1(G). The measure μ will be called isotone on G mapping f→μ*f is isotone, i.e. f≧0 a.e. (m) if and only if μ*f≧0 a.e. (m).

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.

Translation-Invariant Operators On Lp(G), 0 < p < 1 (II)

1977 ◽  
Vol 29 (3) ◽  
pp. 626-630 ◽  
Author(s):  
Daniel M. Oberlin
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Lebesgue Space ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Translation Invariant ◽  
Translation Operators ◽  
Left And Right

For a locally compact group G, let LP(G) be the usual Lebesgue space with respect to left Haar measure m on G. For x ϵ G define the left and right translation operators Lx and Rx by Lx f(y) = f(xy), Rx f(y) = f(yx)(f ϵ Lp(G),y ϵ G). The purpose of this paper is to prove the following theorem.

scholarly journals On Theorems of Kawada and Wendel

1958 ◽  
Vol 11 (2) ◽  
pp. 71-77 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Convolution Product ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Complex Functions ◽  
Image Position ◽  
Locally Compact Topological Group ◽  
Invariant Haar Measure

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li byis again in L1, and

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

scholarly journals The Spectral Radius Formula for Fourier–Stieltjes Algebras

2019 ◽  
Vol 63 (2) ◽  
pp. 269-275
Author(s):  
Przemysław Ohrysko ◽  
Maria Roginskaya
Keyword(s):  
Abelian Group ◽  
Spectral Radius ◽  
Haar Measure ◽  
Short Note ◽  
Measure Algebra ◽  
Stieltjes Transform ◽  
Locally Compact ◽  
Convolution Powers ◽  
Discrete Abelian Group ◽  
Spectral Radius Formula

AbstractIn this short note we first extend the validity of the spectral radius formula, obtained by M. Anoussis and G. Gatzouras, for Fourier–Stieltjes algebras. The second part is devoted to showing that, for the measure algebra on any locally compact non-discrete Abelian group, there are no non-trivial constraints among three quantities: the norm, the spectral radius, and the supremum of the Fourier–Stieltjes transform, even if we restrict our attention to measures with all convolution powers singular with respect to the Haar measure.

On the convergence of iterates of convolution operators in Banach spaces

2020 ◽  
Vol 126 (2) ◽  
pp. 339-366
Author(s):  
Heybetkulu Mustafayev
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Continuous Representation ◽  
Convolution Operators ◽  
Locally Compact ◽  
Bounded Linear ◽  
Almost Everywhere ◽  
Power Bounded

Let $G$ be a locally compact abelian group and let $M(G)$ be the measure algebra of $G$. A measure $\mu \in M(G)$ is said to be power bounded if $\sup _{n\geq 0}\lVert \mu ^{n} \rVert _{1}<\infty $. Let $\mathbf {T} = \{ T_{g}:g\in G\}$ be a bounded and continuous representation of $G$ on a Banach space $X$. For any $\mu \in M(G)$, there is a bounded linear operator on $X$ associated with µ, denoted by $\mathbf {T}_{\mu }$, which integrates $T_{g}$ with respect to µ. In this paper, we study norm and almost everywhere behavior of the sequences $\{ \mathbf {T}_{\mu }^{n}x\}$ $(x\in X)$ in the case when µ is power bounded. Some related problems are also discussed.

Sign in / Sign up

Export Citation Format

Share Document