DTC sets for convolution algebras

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_

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

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.

A discrete norm on an Abelian groupA is a non-negative function ‖ · ‖A → ℝ which satisfies the triangle inequality, is homogenous with respect to scaling ofA by ℤ and is bounded away from 0 onA/{0}. A countable Abelian group is discretely normed if and only if the group is free.

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 study Pontryagin reflexivity in the class of precompact topological Abelian groups. We find reflexive groups among precompact not pseudocompact and among pseudocompact not compact groups. Making use of Martin's axiom we give an example of a reflexive countably compact not compact Abelian group. We also prove that every pseudocompact Abelian group is a quotient of a reflexive pseudocompact group with respect to a closed reflexive pseudocompact subgroup.

In this note we point out two properties (Theorems 1 and 2) of the group algebra of the direct product of a compact group and a locally compact abelian group and indicate briefly how these properties enable one to extend to these group algebras theorems which are known in the compact and abelian cases. Our interest in these properties is two-fold; they establish the applicability of known results in [4]1 to the group algebras in question, and they provide an interesting example of a type of B-algebra, a sort of Kroliecker product algebra, which may very well deserve independenlt systematic study. The reader is referred to results of Kaplansky [1 ] and Segal [2 ] of which this note is essentially a continuation. Let C be a compact group, A be a locally compact abelian group, and G = CXA be their direct product group. We are interested in the group algebras L1(C), L1(A), and L1(G). In particular, we ask how the latter is related to the first two. We will show (1) how the B-algebra L1(G) is formed out of the B-algebras L1(C) and L1(A) and (2) how the structure space of L1(G) is formed from the structure spaces of L1(C) and L1(A). DEFINITION 1. The complex algebra R is the Kronecker product of the complex algebras R1 and R2 (written R = R1 XR2) if there exists a map (xl, X2)>X1 X X2 of (R1, R2) into R such that

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.

Given a compact group G and r ∈ N, let sr(G) denote the least upper bound of real ϵ > 0 such that for every measurable r-coloring of G, there exists a monochrome symmetric subset of measure ≥ ϵ. A subset A ⊆ G is symmetric if there exists g ∈ G such that gA −1 g = A. We give a general picture of asymptotic behaviour of the function sr(G) for compact Abelian groups.

In Pontrjagin's theory of duality for compact abelian groups, the following theorem is known : Let G be a compact abelian group, G* the dual group. Then the topological dimension of G, in the sense of Lebesgue, is equal to the rank of discrete abelian group G*. It was Prof. T. Tannaka who has called my attention to the lack of corresponding theorem in non-commutative case. I intend to give, in this note, a theorem of this kind in the following form: THEOREM A. Let G be an arbitrary compact group, G the aggregate of continuous finite dimensional representstions of G, C[ G ] the algebra over the complex numbers C generated by the coefficients of representations in G, i.e., the representative ring of G in the sense of C. Chevalley^. Then the topological dimension of G, in the sense of Lebesgue, is equal to the transcendental degree of C[G] over C.

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.

To study outer actions $\alpha$ of a group $G$ on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$, $0&lt;\lambda&lt;1$, we study first the cohomology group of a group with the unitary group of an abelian von Neumann algebra as a coefficient group and establish a technique to reduce the coefficient group to the torus $\mathsf T$ by the Shapiro mechanism based on the groupoid approach. We then show a functorial construction of outer actions of a countable discrete amenable group on an AFD factor of type $\mathrm{III}_\lambda$, sharpening the result in [17, §4]. The periodicity of the flow of weights on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$ allows us to introduce an equivariant commutative square directly related to the discrete core. But this makes it necessary to introduce an enlarged group $\mathrm{Aut}(\mathcal M)_{m}$ relative to the modulus homomorphism $m=\mod\colon \mathrm{Aut}(\mathcal M)\to \mathsf R/T'\mathsf Z$. We then discuss the reduced modified HJR-exact sequence, which allows us to describe the invariant of outer action $\alpha$ in a simpler form than the one for a general AFD factor: for example, the cohomology group $H_{m,s}^{out}(G,N,\mathsf T)$ of modular obstructions is a compact abelian group. Making use of these reductions, we prove the classification result of outer actions of $G$ on an AFD factor $\mathcal M$ of type $\mathrm{III}_{\lambda}$.

Given a locally compact group G, G L U C is the largest semigroup compactification of G and G * = G L U C ∖ G . We show that (i) for every locally compact compactly generated Abelian group G and for every p , q ∈ G * , the multiplication in G L U C is discontinuous at ( p , q ) , (ii) there is a locally compact σ-compact torsion-free Abelian group G for which, assuming Martin's Axiom, there are p , q ∈ G * such that the multiplication in G L U C is continuous at ( p , q ) , and (iii) it is consistent with ZFC that for every locally compact Abelian group G and for every p , q ∈ G * , the left translation by p in G * is discontinuous at q.

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

We prove that the category M of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category A of abelian Polish groups in the sense of Beilinson–Bernstein–Deligne and Schneiders. Thus, M is an abelian category containing A as a full subcategory such that the inclusion functor A→M is exact and finitely continuous. Furthermore, M is uniquely characterized up to equivalence by the following universal property: for every abelian category B, a functor A→B is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor M→B. In particular, this provides a description of the left heart of A as a concrete category.We provide similar descriptions of the left heart of a number of categories of algebraic structures endowed with a topology, including: non-Archimedean abelian Polish groups; locally compact abelian Polish groups; totally disconnected locally compact abelian Polish groups; Polish R-modules, for a given Polish group or Polish ring R; and separable Banach spaces and separable Fréchet spaces over a separable complete non-Archimedean valued field.