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

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.

Locally compact groups with totally disconnected space of subgroups

The closed subsets {\mathcal{F}(X)} of any topological space can be given a topology, called the Chabauty–Fell topology, in which {\mathcal{F}(X)} is quasi-compact. If X is a locally compact space, then {\mathcal{F}(X)} is Hausdorff. If {X=G} is a locally compact group, then the subset {\mathcal{S\kern-0.853583ptU\kern-0.569055ptB}(G)} of closed subgroups is closed in {\mathcal{F}(G)} , and the set of closed subgroups inherits its own compact topology, called the Chabauty topology. We develop some of the important properties of this topology. More precisely, the results contained in this paper deal with the following question: Given a locally compact topological group, when is {\mathcal{F}(G)} (resp. {\mathcal{S\kern-0.853583ptU\kern-0.569055ptB}(G)} ) a totally disconnected space?

On topological groups with remainder of character k

<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>

CONTINUITY OF MEASURABLE HOMOMORPHISMS

We 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.

Functional Equations and μ-Spherical Functions

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.

Discrete Sets and Associated Dynamical Systems in a Non-Commutative Setting

We define a uniform structure on the set of discrete sets of a locally compact topological space on which a locally compact topological group acts continuously. Then we investigate the completeness of these uniform spaces and study these spaces by means of topological dynamical systems.

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

Let 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 automorphism group of a connected locally compact topological group

Let G be a locally compact connected topological group. Let Aut0G be the identity component of the group of all bi-continuous automorphisms of G topologized by Birkhoff topology. We give a necessary and sufficient condition for Aut0G to be locally compact.