If $\mu $ is a smooth measure supported on a real-analytic submanifold of ${\mathbb {R}}^{2n}$ which is not contained in any affine hyperplane, then the Weyl transform of $\mu $ is a compact operator.

A suitable notion of weak amenability for dual Banach algebras, which we call weak Connes amenability, is defined and studied. Among other things, it is proved that the measure algebra M(G) of a locally compact group G is always weakly Connes amenable. It can be a complement to Johnson’s theorem that $L^1(G)$ is always weakly amenable [10].

In 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.

Let the measure algebra of a topological group $G$ be equipped with the topology of uniform convergence on bounded right uniformly equicontinuous sets of functions. Convolution is separately continuous on the measure algebra, and it is jointly continuous if and only if $G$ has the $\text{SIN}$ property. On the larger space $\text{LUC}{{(G)}^{*}}$, which includes the measure algebra, convolution is also jointly continuous if and only if the group has the $\text{SIN}$ property, but not separately continuous for many non-SIN groups.

For a Lie group $G$, we show that the map $C_{c}^{\infty }\,\left( G \right)\,\times \,C_{c}^{\infty }\,\left( G \right)\,\to \,C_{c}^{\infty }\,\left( G \right),\,\left( \gamma ,\,\eta \right)\mapsto \,\gamma \,*\,\eta $ taking a pair of test functions to their convolution, is continuous if and only if $G$ is $\sigma -$compact. More generally, consider $r,\,s,\,t\,\in {{\mathbb{N}}_{0}}\,\cup \,\left\{ \infty \right\}$ with $t\,\le \,r\,+\,s$, locally convex spaces ${{E}_{1}}\,,\,{{E}_{2}}$ and a continuous bilinear map $b:\,{{E}_{1}}\,\times \,{{E}_{2}}\,\to \,F$ to a complete locally convex space $F$. Let $\beta :\,C_{c}^{r}\,\left( G,\,{{E}_{1}} \right)\,\times \,C_{c}^{S}\,\left( G,\,{{E}_{2}} \right)\,\to$$C_{c}^{t}\,\left( G,\,F \right),\,\left( \gamma ,\,\eta \right)\,\mapsto \,\gamma \,*\,b\,\eta$ be the associated convolution map. The main result is a characterization of those $\left( G,\,r,s,t,b \right)$ for which $\beta$ is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed as well as convolution of compactly supported ${{L}^{1}}$-functions and convolution of compactly supported Radon measures.

Let ${{A}_{p}}\left( G \right)$ be the Figa-Talamanca, Herz Banach Algebra on $G$; thus ${{A}_{2}}\left( G \right)$ is the Fourier algebra. Strong Ditkin $\left( \text{SD} \right)$ and Extremely Strong Ditkin $\left( \text{ESD} \right)$ sets for the Banach algebras $A_{P}^{r}\left( G \right)$ are investigated for abelian and nonabelian locally compact groups $G$. It is shown that $\text{SD}$ and $\text{ESD}$ sets for ${{A}_{p}}\left( G \right)$ remain $\text{SD}$ and $\text{ESD}$ sets for $A_{P}^{r}\left( G \right)$, with strict inclusion for $\text{ESD}$ sets. The case for the strict inclusion of $\text{SD}$ sets is left open.

A result on the weak sequential completeness of ${{A}_{2}}\left( F \right)$ for $\text{ESD}$ sets $F$ is proved and used to show that Varopoulos, Helson, and Sidon sets are not $\text{ESD}$ sets for ${{A}_{2}}\left( G \right)$, yet they are such for $A_{2}^{r}\left( G \right)$ for discrete groups $G$, for any $1\,\le \,r\,\le \,2$.

A result is given on the equivalence of the sequential and the net definitions of $\text{SD}$ or $\text{ESD}$ sets for $\sigma $-compact groups.

The above results are new even if $G$ is abelian.

In this article we study the Fourier space of a general hypergroup and its multipliers. The main result of this paper characterizes commutative hypergroups whose Fourier space forms a Banach algebra under pointwise product with an equivalent norm. Among those hypergroups whose Fourier space forms a Banach algebra, we identify a subclass for which the Gelfand spectrum of the Fourier algebra is equal to the underlying hypergroup. This subclass includes for instance, Jacobi hypergroups, Bessel-Kingman hypergroups.

Let φ be a continuous nonzero homomorphism of the convolution algebra L1loc(R+) and also the unique extension of this homomorphism to Mloc(R+). We show that the map φis continuous in the weak* and strong opertor topologies on Mloc, considered as the dual space of Cc(R+) and as the multiplier algebra of L1loc. Analogous results are proved for homomorphism from L1 [0, a) to L1 [0, b). For each convolution algebra L1 (ω1), φ restricts to a continuous homomorphism from some L1 (ω1) to some L1 (ω2), and, for each sufficiently large L1 (ω2), φ restricts to a continuous homomorphism from some L1 (ω1) to L1 (ω2). We also determine which continuous homomorphisms between weighted convolution algebras extend to homomorphisms of L1loc. We also prove results on convergent nets, continuous semigroups, and bounded sets in Mloc that we need in our study of homomorphisms.

We generalise group algebras to other algebraic objects with bounded Hilbert space representation theory; the generalised group algebras are called ‘host’ algebras. The main property of a host algebra is that its representation theory should be isomorphic (in the sense of the Gelfand–Raikov theorem) to a specified subset of representations of the algebraic object. Here we obtain both existence and uniqueness theorems for host algebras as well as general structure theorems for host algebras. Abstractly, this solves the question of when a set of Hilbert space representations is isomorphic to the representation theory of a C*-algebra. To make contact with harmonic analysis, we consider general convolution algebras associated to representation sets, and consider conditions for a convolution algebra to be a host algebra.

We study convergence in weighted convolution algebras ${{L}^{1}}\left( \omega \right)$ on ${{R}^{+}}$, with the weights chosen such that the corresponding weighted space $M\left( \omega \right)$ of measures is also a Banach algebra and is the dual space of a natural related space of continuous functions. We determine convergence factor $\eta $ for which weak$*$-convergence of $\left\{ {{\text{ }\!\!\lambda\!\!\text{ }}_{n}} \right\}$ to $\text{ }\!\!\lambda\!\!\text{ }$ in $M\left( \omega \right)$ implies norm convergence of ${{\text{ }\!\!\lambda\!\!\text{ }}_{n}}*f$ to $\text{ }\!\!\lambda\!\!\text{ *}f$ in ${{L}^{1}}\left( \omega \eta \right)$. We find necessary and sufficent conditions which depend on $\omega$ and $f$ and also find necessary and sufficent conditions for $\eta$ to be a convergence factor for all ${{L}^{1}}\left( \omega \right)$ and all $f$ in ${{L}^{1}}\left( \omega \right)$. We also give some applications to the structure of weighted convolution algebras. As a preliminary result we observe that $\eta$ is a convergence factor for $\omega$ and $f$ if and only if convolution by $f$ is a compact operator from $M\left( \omega \right)$ (or ${{L}^{1}}\left( \omega \right)$) to ${{L}^{1}}\left( \omega \eta \right)$.

In this paper we make use of semigroup methods on the space of compactly supported measures to obtain a Bochner representation for α-bounded positive-definite functions on a commutative hypergroup.

A compact hypergroup is called almost discrete if it is homeomorphic to the one-point-compactification of a countably infinite discrete set. If the group Up of all p-adic units acts multiplicatively on the p-adic integers, then the associated compact orbit hypergroup has this property. In this paper we start with an exact projective sequence of finite hypergroups and use successive substitution to construct a new surjective projective system of finite hypergroups whose limit is almost discrete. We prove that all compact almost discrete hypergroups appear in this way—up to isomorphism and up to a technical restriction. We also determine the duals of these hypergroups, and we present some examples coming from partitions of compact totally disconnected groups.

In this paper we make use of semigroup methods on the space of compactly supported probability measures to obtain a complete Lévy-Khinchin representation for negative definite functions on a commutative hypergroup. In addition we obtain representation theorems for completely monotone and completely alternating functions. The techniques employed here also lead to considerable simplification of the proofs of known results on positive definite and negative definite functions on hypergroups.

It is shown that suitable conditions on the components of a measure in the algebra M(G) ensure that this element lies outside of all maximal bi-ideals.

Let S and T be locally compact topological semigroups and a semidirect product. Conditions are determined under which topological left amenability of S and T implies that of , and conversely. The results are used to show that for a large class of semigroups which are neither compact nor groups, various notions of topological left amenability coincide.