If G is a locally compact group with a compact invariant neighbourhood of the identity e, the following property (*) holds: For every continuous positive definite function h≥ 0 with compact support there is a constant $C_h>0$ such that $\int L_{x}h·g\le C_h\int hg$ for every continuous positive definite g≥0, where $L_x$ is left translation by x. In [L], property (*) was stated, but the above inequality was proved for special h only. That “for one h” implies “for all h” seemed obvious, but turned out not to be obvious at all. We fill...

Let $\Gamma$ be an irreducible lattice in a product $G$ of simple groups. Assume that $G$ has a factor with property (T). We give a description of the topology in a neighbourhood of the trivial one dimensional representation of $\Gamma$ in terms of the topology of the dual space $\widehat{G}$ of $G$.We use this result to give a new proof for the triviality of the first cohomology group of $\Gamma$ with coefficients in a finite dimensional unitary representation.

We show how the measure theory of regular compacted-Borel measures defined on the $\delta$-ring of compacted-Borel subsets of a weighted locally compact group $(G,\omega )$ provides a compatible framework for defining the corresponding Beurling measure algebra $ℳ(G,\omega )$, thus filling a gap in the literature.

We obtain characterizations of left character amenable Banach algebras in terms of the existence of left ϕ-approximate diagonals and left ϕ-virtual diagonals. We introduce the left character amenability constant and find this constant for some Banach algebras. For all locally compact groups G, we show that the Fourier-Stieltjes algebra B(G) is C-character amenable with C < 2 if and only if G is compact. We prove that if A is a character amenable, reflexive, commutative Banach algebra, then A...

Let G be a locally compact Polish group with an invariant metric. We provide sufficient and necessary conditions for the existence of a compact set A ⊆ G and a sequence $g_n\in G$ such that ${\mu }^{\ast n}\left(g_{n}A\right)\equiv 1$ for all n. It is noticed that such measures μ form a meager subset of all probabilities on G in the weak measure topology. If for some k the convolution power ${\mu }^{\ast k}$ has nontrivial absolutely continuous component then a similar characterization is obtained for any locally compact, σ-compact, unimodular, Hausdorff topological...

Let $G$ be a Polish group with an invariant metric. We characterize those probability measures $\mu$ on $G$ so that there exist a sequence $g_n\in G$ and a compact set $A\subseteq G$ with   ${\mu }^{*n}\left(g_{n}A\right)\equiv 1$   for all $n$.

Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space ℒ(H) of bounded linear operators on H with the weak operator topology. We prove that if U is a measurable map from G to ℒ(H) then it is continuous. This result was known before for separable H. We also prove that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous.

Let $C{V}_p\left(F\right)$ be the left convolution operators on $L^p\left(G\right)$ with support included in F and $M_p\left(F\right)$ denote those which are norm limits of convolution by bounded measures in M(F). Conditions on F are given which insure that $C{V}_p\left(F\right)$, $C{V}_p\left(F\right)/M_p\left(F\right)$ and $C{V}_p\left(F\right)/W$ are as big as they can be, namely have $l^{\infty }$ as a quotient, where the ergodic space W contains, and at times is very big relative to $M_p\left(F\right)$. Other subspaces of $C{V}_p\left(F\right)$ are considered. These improve results of Cowling and Fournier, Price and Edwards, Lust-Piquard, and others.

The convolution of ultrafilters of closed subsets of a normal topological group is considered as a substitute of the extension onto ${\left(\beta \right)}^2$ of the group operation. We find a subclass of ultrafilters for which this extension is well-defined and give some examples of pathologies. Next, for a given locally compact group and its dense subgroup , we construct subsets of β algebraically isomorphic to . Finally, we check whether the natural mapping from β onto β is a homomorphism with respect to the extension...