### A Generalized Orlicz-Pettis Theorem and Applications.

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The study of Gaussian convolution semigroups is a subject at the crossroad between abstract and concrete problems in harmonic analysis. This article suggests selected open problems that are in large part motivated by joint work with Alexander Bendikov.

We calculate the cardinal characteristics of the $\sigma $-ideal $\mathscr{H}\mathcal{N}\left(G\right)$ of Haar null subsets of a Polish non-locally compact group $G$ with invariant metric and show that $cov\left(\mathscr{H}\mathcal{N}\right(G\left)\right)\le \U0001d51f\le max\{\U0001d521,non(\mathcal{N}\left)\right\}\le non\left(\mathscr{H}\mathcal{N}\right(G\left)\right)\le cof\left(\mathscr{H}\mathcal{N}\right(G\left)\right)\phantom{\rule{-0.86pt}{0ex}}>\phantom{\rule{-0.86pt}{0ex}}min\{\U0001d521,non(\mathcal{N}\left)\right\}$. If $G={\prod}_{n\ge 0}{G}_{n}$ is the product of abelian locally compact groups ${G}_{n}$, then $add\left(\mathscr{H}\mathcal{N}\right(G\left)\right)=add\left(\mathcal{N}\right)$, $cov\left(\mathscr{H}\mathcal{N}\right(G\left)\right)=min\{\U0001d51f,cov(\mathcal{N}\left)\right\}$, $non\left(\mathscr{H}\mathcal{N}\right(G\left)\right)=max\{\U0001d521,non(\mathcal{N}\left)\right\}$ and $cof\left(\mathscr{H}\mathcal{N}\right(G\left)\right)\ge cof\left(\mathcal{N}\right)$, where $\mathcal{N}$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $cof\left(\mathscr{H}\mathcal{N}\left(G\right)\right)>{2}^{{\aleph}_{0}}$ and hence $G$ contains a Haar null subset that cannot be enlarged to a Borel or projective Haar null subset of $G$. This gives a negative (consistent) answer to a question of...

We give a review of results proved and published mostly in recent years, concerning real-valued convex functions as well as almost convex functions defined on a (not necessarily convex) subset of a group. Analogues of such classical results as the theorems of Jensen, Bernstein-Doetsch, Blumberg-Sierpiński, Ostrowski, and Mehdi are presented. A version of the Hahn-Banach theorem with a convex control function is proved, too. We also study some questions specific for the group setting, for instance...

We prove that the topology of the additive group of the Banach space c₀ is not induced by weakly almost periodic functions or, what is the same, that this group cannot be represented as a group of isometries of a reflexive Banach space. We show, in contrast, that additive groups of Schwartz locally convex spaces are always representable as groups of isometries on some reflexive Banach space.

Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ${(G,\tau )}^{\wedge}$ is a dense -closed subgroup of the compact group ${\left({G}_{d}\right)}^{\wedge}$, where ${G}_{d}$ is the group G with the discrete...

Let $X$ be an Abelian topological group. A subgroup $H$ of $X$ is characterized if there is a sequence $\mathbf{u}=\left\{{u}_{n}\right\}$ in the dual group of $X$ such that $H=\{x\in X:\phantom{\rule{0.277778em}{0ex}}({u}_{n},x)\to 1\}$. We reduce the study of characterized subgroups of $X$ to the study of characterized subgroups of compact metrizable Abelian groups. Let ${c}_{0}\left(X\right)$ be the group of all $X$-valued null sequences and ${\U0001d532}_{0}$ be the uniform topology on ${c}_{0}\left(X\right)$. If $X$ is compact we prove that ${c}_{0}\left(X\right)$ is a characterized subgroup of ${X}^{\mathbb{N}}$ if and only if $X\cong {\mathbb{T}}^{n}\times F$, where $n\ge 0$ and $F$ is a finite Abelian group. For every compact Abelian...