Functional Equations and μ-Spherical Functions

2008 ◽  
Vol 15 (1) ◽  
pp. 1-20
Author(s):  
Mohamed Akkouchi ◽  
Belaid Bouikhalene ◽  
Elhoucien Elqorachi
Keyword(s):  
Functional Equation ◽  
Topological Group ◽  
Functional Equations ◽  
Compact Subgroup ◽  
Aequationes Math ◽  
Gelfand Pair ◽  
Locally Compact ◽  
The Matrix ◽  
The One ◽  
Locally Compact Topological Group

Abstract We will study the properties of solutions 𝑓, {𝑔𝑖}, {ℎ𝑖} ∈ 𝐶𝑏(𝐺) of the functional equation where 𝐺 is a Hausdorff locally compact topological group, 𝐾 a compact subgroup of morphisms of 𝐺, χ a character on 𝐾, and μ a 𝐾-invariant measure on 𝐺. This equation provides a common generalization of many functional equations (D'Alembert's, Badora's, Cauchy's, Gajda's, Stetkaer's, Wilson's equations) on groups. First we obtain the solutions of Badora's equation [Aequationes Math. 43: 72–89, 1992] under the condition that (𝐺,𝐾) is a Gelfand pair. This result completes the one obtained in [Badora, Aequationes Math. 43: 72–89, 1992] and [Elqorachi, Akkouchi, Bakali and Bouikhalene, Georgian Math. J. 11: 449–466, 2004]. Then we point out some of the relations of the general equation to the matrix Badora functional equation and obtain explicit solution formulas of the equation in question for some particular cases. The results presented in this paper may be viewed as a continuation and a generalization of Stetkær's, Badora's, and the authors' works.

scholarly journals CONTINUITY OF MEASURABLE HOMOMORPHISMS

2008 ◽  
Vol 78 (1) ◽  
pp. 171-176 ◽  
Author(s):  
JANUSZ BRZDȨK
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Baire Property ◽  
Topological Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Locally Compact Topological Group

AbstractWe give some general results concerning continuity of measurable homomorphisms of topological groups. As a consequence we show that a Christensen measurable homomorphism of a Polish abelian group into a locally compact topological group is continuous. We also obtain similar results for the universally measurable homomorphisms and the homomorphisms that have the Baire property.

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

scholarly journals On topological groups with remainder of character k

2016 ◽  
Vol 17 (1) ◽  
pp. 51
Author(s):  
Maddalena Bonanzinga ◽  
Maria Vittoria Cuzzupè
Keyword(s):  
Topological Group ◽  
Infinite Cardinal ◽  
Topological Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Locally Compact Topological Group

<p style="margin: 0px;">In [A.V. Arhangel'skii and J. van Mill, On topological groups with a first-countable remainder, Top. Proc. <span id="OBJ_PREFIX_DWT1099_com_zimbra_phone" class="Object">42 (2013), 157-163</span>] it is proved that the character of a non-locally compact topological group with a first countable remainder doesn't exceed $\omega_1$ and a non-locally compact topological group of character $\omega_1$ having a compactification whose reminder is first countable is given. We generalize these results in the general case of an arbitrary infinite cardinal k.</p><p style="margin: 0px;"> </p>

Actions of a Locally Compact Group with Zero

1971 ◽  
Vol 23 (3) ◽  
pp. 413-420 ◽  
Author(s):  
T. H. McH. Hanson
Keyword(s):  
Topological Group ◽  
Compact Group ◽  
Topological Semigroup ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Definition Of ◽  
Isolated Point ◽  
Locally Compact Topological Group

In [2] we find the definition of a locally compact group with zero as a locally compact Hausdorff topological semigroup, S, which contains a non-isolated point, 0, such that G = S – {0} is a group. Hofmann shows in [2] that 0 is indeed a zero for S, G is a locally compact topological group, and the unit, 1, of G is the unit of S. We are to study actions of S and G on spaces, and the reader is referred to [4] for the terminology of actions.If X is a space (all are assumed Hausdorff) and A ⊂ X, A* denotes the closure of A. If {xρ} is a net in X, we say limρxρ = ∞ in X if {xρ} has no subnet which converges in X.

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.

On the supports of Gauss measures on algebraic groups

1984 ◽  
Vol 96 (3) ◽  
pp. 437-445 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Topological Semigroup ◽  
Weak Topology ◽  
Algebraic Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Borel Measures ◽  
Image Position ◽  
Locally Compact Topological Group

For any locally compact topological group G let M(G) denote the topological semigroup of all probability (Borel) measures on G, furnished with the weak topology and with convolution as the multiplication. A Gauss semigroup on G is a homomorphism t→ μt of the strictly positive reals (under addition) into M(G) such that(i) no μt is a point mesaure,(ii) for each neighbourhood V of 1 in G we have

scholarly journals On One-parameter Subgroups in Finite Dimensional Locally Compact Group with No Small Subgroups

1952 ◽  
Vol 4 ◽  
pp. 89-96
Author(s):  
Masatake Kuranishi
Keyword(s):  
Topological Group ◽  
Compact Group ◽  
Lie Group ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Finite Dimensional ◽  
Locally Compact Topological Group

Let G be a locally compact topological group and let U be a neighborhood of the identity in G. A curve g(λ) (|λ| ≦ 1) in G, which satisfies the conditions, g(s)g(t) = g(s + t) (|s|, |f|, |s + t| ≦ l),is called a one-parameter subgroup of G. If there exists a neighborhood U1 of the identity in G such that for every element x of U1 there exists a unique one-parameter subgroup g(λ) which is contained in U and g(1) =x, we shall call, for the sake of simplicity, that U has the property (S). It is well known that the neighborhoods of the identity in a Lie group have the property (S). More generally it is proved that if G is finite dimensional, locally connected, and is without small subgroups, G has the same property. In this note, these theorems will be generalized to the case when G is unite dimensional and without small subgroups.

scholarly journals On the Baum–Connes Conjecture for Groups Acting on CAT(0)-Cubical Spaces

2020 ◽  
Author(s):  
Jacek Brodzki ◽  
Erik Guentner ◽  
Nigel Higson ◽  
Shintaro Nishikawa
Keyword(s):  
Topological Group ◽  
Splitting Method ◽  
Locally Compact ◽  
Bounded Geometry ◽  
Compact Topological Group ◽  
Finite Dimensional ◽  
Locally Compact Topological Group ◽  
Kasparov Theory

Abstract We give a new proof of the Baum–Connes conjecture with coefficients for any second countable, locally compact topological group that acts properly and cocompactly on a finite-dimensional CAT(0)-cubical space with bounded geometry. The proof uses the Julg–Valette complex of a CAT(0)-cubical space introduced by the 1st three authors and the direct splitting method in Kasparov theory developed by the last author.

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