Plünnecke inequalities for countable abelian groups

We establish in this paper a new form of Plünnecke-type inequalities for ergodic probability measure-preserving actions of any countable abelian group. Using a correspondence principle for product sets, this allows us to deduce lower bounds on the upper and lower Banach densities of any product set in terms of the upper Banach density of an iterated product set of one of its addends. These bounds are new already in the case of the integers.We also introduce the notion of an ergodic basis, which is parallel, but significantly weaker than the analogous notion of an additive basis, and deduce Plünnecke bounds on their impact functions with respect to both the upper and lower Banach densities on any countable abelian group.

Essential Commutants of Semicrossed Products

Let α: G ↷ M be a spatial action of a countable abelian group on a “spatial” von Neumann algebra M and let S be its unital subsemigroup with G = S-1S. We explicitly compute the essential commutant and the essential fixed-points, modulo the Schatten p-class or the compact operators, of the w*-semicrossed product of M by S when M' contains no non-zero compact operators. We also prove a weaker result when M is a von Neumann algebra on a finite dimensional Hilbert space and (G, S) = (ℤ, ℤ+), which extends a famous result due to Davidson (1977) for the classical analytic Toeplitz operators.

Joint ergodicity of actions of an abelian group

Let $G$ be a countable abelian group and let ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ be measure preserving $G$-actions on a probability space. We prove that joint ergodicity of ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ implies total joint ergodicity if each ${T}^{(i)} $ is totally ergodic. We also show that if $G= { \mathbb{Z} }^{d} $, $s\geq d+ 1$ and the actions ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ commute, then total joint ergodicity of ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ follows from joint ergodicity. This can be seen as a generalization of Berend’s result for commuting $ \mathbb{Z} $-actions.

Geometric K-homology with coefficients I: ℤ/kℤ-cycles and Bockstein sequence

We construct a Baum-Douglas type model for K-homology with coefficients in ℤ/kℤ. The basic geometric object in a cycle is a spinc ℤ/kℤ-manifold. The relationship between these cycles and the topological side of the Freed-Melrose index theorem is discussed in detail. Finally, using inductive limits, we construct geometric models for K-homology with coefficients in any countable abelian group.

Frames and Single Wavelets for Unitary Groups

We consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson’s theorem to apply to frames generated by the action of the group. Within this setup we use Stone’s theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multiresolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on L2(ℝ).

Multiple Mixing and Rank One Group Actions

Suppose G is a countable, Abelian group with an element of infinite order and let be amixing rank one action of G on a probability space. Suppose further that the Følner sequence {Fn} indexing the towers of satisfies a “bounded intersection property”: there is a constant p such that each {Fn} can intersect no more than p disjoint translates of {Fn}. Then is mixing of all orders. When G = Z, this extends the results of Kalikow and Ryzhikov to a large class of “funny” rank one transformations. We follow Ryzhikov’s joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of k copies of is necessarily product measure. This method generalizes Ryzhikov’s technique.

Morse cocycles and simple Lebesgue spectrum

It is well known that the spectrum of dynamical systems arising from generalized Morse sequences (M. Keane. Generalized Morse sequences. Z. Wahr. Verw. Geb.10 (1968), 335–353.) is simple as soon as it is non-discrete. We give, in this paper, a necessary and sufficient condition for the existence of such a transformation with non-purely singular spectrum. From this, it follows that this problem is equivalent to an open problem of the existence of ‘flat’ polynomials on the Torus group. We show that this latter question can be given an affirmative answer on some other group, and this allows us to construct a countable abelian group action with simple spectrum whose spectral type is the sum of a discrete measure and of the Haar measure on the dual group.

Mixing properties of induced random transformations

Let $S(N)$ be a random walk on a countable abelian group $G$ which acts on a probability space $E$ by measure-preserving transformations $(T_v)_{v\in G}$. For any $\Lambda \subset E$ we consider the random return time $\tau$ at which $T_{S(\tau)}\in\Lambda$. We show that the corresponding induced skew product transformation is K-mixing whenever a natural subgroup of $G$ acts ergodically on $E$.