Various proofs of the Factorization Theorem for representations of Banach algebras are compared with its original proof due to P. Cohen.

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 P(z,β) be the Poisson kernel in the unit disk , and let $P_{\lambda }f\left(z\right)={ʃ}_{\partial }P{(z,\phi )}^{1/2+\lambda }f\left(\phi \right)d\phi$ be the λ -Poisson integral of f, where $f\in L^p\left(\partial \right)$. We let $P_{\lambda }f$ be the normalization $P_{\lambda }f/P_{\lambda }1$. If λ >0, we know that the best (regular) regions where $P_{\lambda }f$ converges to f for a.a. points on ∂ are of nontangential type. If λ =0 the situation is different. In a previous paper, we proved a result concerning the convergence of $P_{0}f$ toward f in an $L^p$ weakly tangential region, if $f\in L^p\left(\partial \right)$ and p > 1. In the present paper we will extend the result to symmetric spaces X of...

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.

We prove that for every compact, connected group G there is a singular measure μ such that the Fourier series of μ*μ converges uniformly on G. Our results extend the earlier results of Saeki and Dooley-Gupta.

The convolution kernels $Vf=f*x$ on a homogeneous space $E=G/K$, where $K$ is a compact sub-group of $G$, that satisfy the complete maximum principle are characterized.Deny’s result for abelian groups $G$, but with a stronger hypothesis, is a special case.

We discuss some results about derivations and crossed homomorphisms arising in the context of locally compact groups and their group algebras, in particular, L¹(G), the von Neumann algebra VN(G) and actions of G on related algebras. We answer a question of Dales, Ghahramani, Grønbæk, showing that L¹(G) is always permanently weakly amenable. Then we show that for some classes of groups (e.g. IN-groups) the homology of L¹(G) with coefficients in VN(G) is trivial. But this is no longer true, in general,...

In the study of spectral synthesis S-sets and C-sets (see Rudin [3]; Reiter [2] uses the terminology Wiener sets and Wiener-Ditkin sets respectively) have been discussed extensively. A new concept of Ditkin sets was introduced and studied by Stegeman in [4] so that, in Reiter’s terminology, Wiener-Ditkin sets are precisely sets which are both Wiener sets and Ditkin sets. The importance of such sets in spectral synthesis and their connection to the C-set-S-set problem (see Rudin [3]) are mentioned...