### The Borel Property for Simple Riesz Means.

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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.

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,...

We consider multi-dimensional Hartman almost periodic functions and sequences, defined with respect to different averaging sequences of subsets in ${\mathbb{R}}^{d}$ or ${\mathbb{Z}}^{d}$. We consider the behavior of their Fourier-Bohr coefficients and their spectrum, depending on the particular averaging sequence, and we demonstrate this dependence by several examples. Extensions to compactly generated, locally compact, abelian groups are considered. We define generalized Marcinkiewicz spaces based upon arbitrary measure spaces...

Let ${X}_{i}$, i∈ I, and ${Y}_{j}$, j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of ${\prod}_{i\in I}{X}_{i}$ onto ${\prod}_{j\in J}{Y}_{j}$. We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of ${X}_{i}$ onto ${Y}_{b\left(i\right)}$, i∈ I.

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