### A characterisation of the circle group.

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For a locally compact, abelian group $G$, we study the space ${S}_{0}\left(G\right)$ of functions on $G$ belonging locally to the Fourier algebra and with ${l}^{1}$-behavior at infinity. We give an abstract characterization of the family of spaces $\left\{{S}_{0}\right(G):G$ abelian$\}$ by its hereditary properties.

In this paper we seek to describe the structure of self-dual torsion-free LCA groups. We first present a proof of the structure theorem of self-dual torsion-free metric LCA groups. Then we generalize the structure theorem to a larger class of self-dual torsion-free LCA groups. We also give a characterization of torsion-free divisible LCA groups. Consequently, a complete classification of self-dual divisible LCA groups is obtained; and any self-dual torsion-free LCA group can be regarded as an open...

The existence of a projection onto an ideal I of a commutative group algebra ${L}^{1}\left(G\right)$ depends on its hull Z(I) ⊆ Ĝ. Existing methods for constructing a projection onto I rely on a decomposition of Z(I) into simpler hulls, which are then reassembled one at a time, resulting in a chain of projections which can be composed to give a projection onto I. These methods are refined and examples are constructed to show that this approach does not work in general. Some answers are also given to previously asked...

We prove that every connected locally compact Abelian topological group is sequentially connected, i.e., it cannot be the union of two proper disjoint sequentially closed subsets. This fact is then applied to the study of extensions of topological groups. We show, in particular, that if $H$ is a connected locally compact Abelian subgroup of a Hausdorff topological group $G$ and the quotient space $G/H$ is sequentially connected, then so is $G$.

Gruenhage asked if it was possible to cover the real line by less than continuum many translates of a compact nullset. Under the Continuum Hypothesis the answer is obviously negative. Elekes and Stepr mans gave an affirmative answer by showing that if ${C}_{EK}$ is the well known compact nullset considered first by Erdős and Kakutani then ℝ can be covered by cof() many translates of ${C}_{EK}$. As this set has no analogue in more general groups, it was asked by Elekes and Stepr mans whether such a result holds for...