Discretely normed Abelian groups

Let K⊂ℝ d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in KΛ, equipped with the product topology, where each coordinate solves a SDE of the form dX i (t) = ∑ j a(j−i) (X j (t) −X i (t))dt + σ (X i (t))dB i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a S (i) = a(i) + a(−i) is recurrent, then each component X i (∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a S on Abelian groups Λ.

We make the following three observations regarding a question popularized by Katznelson: is every subset of \({\mathbb {Z}}\) which is a set of Bohr recurrence also a set of topological recurrence? (i) If G is a countable abelian group and \(E\subseteq G\) is an \(I_0\) set, then every subset of \(E-E\) which is a set of Bohr recurrence is also a set of topological recurrence. In particular every subset of \(\{2^n-2^m : n,m\in {\mathbb {N}}\}\) which is a set of Bohr recurrence is a set of topological recurrence. (ii) Let \({\mathbb {Z}}^{\omega }\) be the direct sum of countably many copies of \({\mathbb {Z}}\) with standard basis E. If every subset of \((E-E)-(E-E)\) which is a set of Bohr recurrence is also a set of topological recurrence, then every subset of every countable abelian group which is a set of Bohr recurrence is also a set of topological recurrence. (iii) Fix a prime p and let \({\mathbb {F}}_p^\omega \) be the direct sum of countably many copies of \({\mathbb {Z}}/p{\mathbb {Z}}\) with basis \(({\mathbf {e}}_i)_{i\in {\mathbb {N}}}\). If for every p-uniform hypergraph with vertex set \({\mathbb {N}}\) and edge set \({\mathcal {F}}\) having infinite chromatic number, the Cayley graph on \(\mathbb F_p^\omega \) determined by \(\{\sum _{i\in F}{\mathbf {e}}_i:F\in {\mathcal {F}}\}\) has infinite chromatic number, then every subset of \(\mathbb F_p^\omega \) which is a set of Bohr recurrence is a set of topological recurrence.

Let $T$ be a measure-preserving and mixing action of a countable abelian group $G$ on a probability space $(X,\mathscr S,\mu )$ and $A$ a locally compact second countable abelian group. A cocycle $c\colon G\times X\to A$ for $T$ <i>disperses</i> if $\lim_

1. Definitions and notations. Let Y be a compact connected metric abelian (c.c.m.a.) group. We shall use additive notation in such groups. Y will denote the discrete torsion-free countable abelian character group of Y, and multiplicative notation will be used in Y. Y can be written as an inverse limit inv lim (Yi,' r), where each Ym (m > 1) is a finite-dimensional torus and grn is a homomorphism of YM+1 onto Ym. If Y is n-dimensional then each Ym can be chosen to be an n

The splitting length of a mixed Abelian group G is defined as the smallest positive integer n such that \( \mathop \otimes \limits^n G \) splits. The task of determining the splitting length of mixed Abelian groups was formulated by Irwin, Khabbaz, and Rayna. In this paper, a criterion for determining whether \( \mathop \otimes \limits^n G \) splits for countable mixed Abelian groups G of torsion-free rank 1 is found.

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

An abstract sampling theory associated with a unitary representation of a countable discrete non abelian group G, which is a semi-direct product of groups, on a separable Hilbert space is studied. A suitable expression of the data samples, the use of a filter bank formalism and the corresponding frame analysis allow for fixing the mathematical problem to be solved: the search of appropriate dual frames for ℓ 2 ( G ) . An example involving crystallographic groups illustrates the obtained results by using either average or pointwise samples.

Let G G be a locally compact SIN group, and w w a weight on G G which is diagonally bounded with bound K K on a dispersed set. We show that the topological centre of L U C ( w − 1 ) ∗ {LUC}(w^{-1})^* equals M ( w ) M(w) , and there are ⌊ K ⌋ + 1 \lfloor K \rfloor + 1 many points in the spectrum of L U C ( w − 1 ) {LUC}(w^{-1}) which are determining for the topological centre (DTC). As corollaries, we obtain DTC results as well for L 1 ( w ) ∗ ∗ L_1(w)^{**} , C 0 ( w − 1 ) ⊥ C_0(w^{-1})^\perp and L ∞ , 0 ( w − 1 ) ⊥ L_{\infty , 0}(w^{-1})^\perp . We also give a short proof of the first result in the unweighted case for all locally compact second countable abelian groups G G .

Nous établissons que, pour tout groupe dénombrable abélien Γ et tout sous-groupe finiment engendré Δ de Γ, l’ensemble des actions de Δ sur un espace de probabilités standard (X,μ) qui peuvent être étendues en une action libre de Γ sur (X,μ) est générique (au sens de Baire). Ce résultat étend un théorème d’Ageev, qui correspond au cas où Δ est un groupe cyclique infini.

<p style='text-indent:20px;'>In this article we study the centralizer of a minimal aperiodic action of a countable group on the Cantor set (an aperiodic minimal Cantor system). We show that any countable residually finite group is the subgroup of the centralizer of some minimal <inline-formula><tex-math id="M1">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula> action on the Cantor set, and that any countable group is the subgroup of the normalizer of a minimal aperiodic action of an abelian countable free group on the Cantor set. On the other hand we show that for any countable group <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>, the centralizer of any minimal aperiodic <inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-action on the Cantor set is a subgroup of the centralizer of a minimal <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Z $\end{document}</tex-math></inline-formula>-action.

In a paper entitled cocycles and spectra Helson and Par ry prove that for every T there exists a q~ such that V ~ has Lebesgue spectrum, moreover q~ can be chosen to be real, i.e., taking values + 1 and 1. The purpose of this paper is to extend this result to certain actions of countable groups which includes ergodic non-singular actions of countable abelian groups. We blend the method of Helson and Parry with the notions of weak equivalence and weak von Neumann transformations. In section 3 and 4 we discuss these results in connection with systems of imprimitivity. The problem of extending the result quoted above to countable groups was raised by H. Helson to one of us. It is a pleasure to acknowledge his interest and encouragement in this work.

Let A be the AF algebra whose scaled ordered group K0(A) is (G⊕H,(G+∖{0})⊕H∪{(0,0)},g˜⊕0), where (G,G+,g˜) is the scaled ordered group K0(B) of a unital simple AF algebra B, and H is a countable torsion-free Abelian group. Let σ be an order 2 scaled ordered automorphism of K0(A), defined by σ(g,h)=(g,−h), where (g,h)∈G⊕H. We show that there is an order 2 automorphism α of A such that α∗=σ. This gives a partial answer to a lifting question posed by Blackadar. Incidentally, the lift α we construct has the tracial Rokhlin property. Consequently, the crossed product C∗(Z2,A,α) is a unital simple AH algebra with no dimension growth.

We study homoclinic points of non-expansive automorphisms of compact abelian groups. Connections between the existence of non-trivial homoclinic points, expansiveness, entropy and adjoint automorphisms (in the sense of Einsiedler and Schmidt) are explored. Some implications for countable abelian group actions by automorphisms of compact abelian groups are also considered and it is shown that if every element of such an action has finite entropy, there can be no non-trivial common homoclinic points, unless the action is generated by a single automorphism.

On asymptotic dimension of countable Abelian groups

If a finite group A of exponent 4 is the automorphism group of a torsionfree (abelian) group G, then A must be a subdirect product of cyclic groups C2 and C4 and of quaternion groups Qs (see [2, 4]); A is therefore contained in the variety V(Qs) generated by the quaternion group. Having dealt completely in [3] with the case of finite (or countable) abelian automorphism groups, we now turn to non-abelian groups (which, as we have mentioned before ([3], p. 34), must be of exponent 4 or 12). In this paper we confine ourselves to proving that for every n>2 the free n-generator group A, of V(Qs) is, in fact, the automorphism group of a suitably chosen torsionfree group 1. We also give at the end of the paper an example (with n = 3) of an epimorphic image of Aa of order 27 that cannot be an automorphism group, although it still satisfies the condition N3 of [2]: all its elements of order 2 lie in its centre. This group fails to satisfy the necessary condition N5 of [1] and cannot be a subdirect product of the kind described above.