Folner Condition Research Articles

Similarity Problems, Følner Sets and Isometric Representations of Amenable Semigroups

We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier-Day theorem concerning bounded representations of amenable groups and semigroups. We highlight the role of Folner sets in similarity problems in both settings of unimodular, $\sigma$-compact, amenable groups and in discrete semigroups possessing the Strong Folner condition (SFC).

Amenability and geometry of semigroups

We study the connection between amenability, Folner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or two-sided zero element), left amenability coincides with the strong Folner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable if and only if it is left reversible. We show that the (weak) Folner condition is a left quasi-isometry invariant of finitely generated semigroups, and hence that left amenability is a left quasi-isometry invariant of left cancellative semigroups. We also give a new characterisation of the strong Folner condition in terms of the existence of weak Folner sets satisfying a local injectivity condition on the relevant translation action of the semigroup.

Minimal Følner foliations are amenable

For finitely generated groups, amenability and Folner properties are equivalent. However, contrary to a widespread idea, Kaimanovich showed that Folner condition does not imply amenability for discrete measured equivalence relations. In this paper, we exhibit two examples of $C^\infty$ foliations of closed manifolds that are Folner and non amenable with respect to a finite transverse invariant measure and a transverse invariant volume, respectively. We also prove the equivalence between the two notions when the foliation is minimal, that is all the leaves are dense, giving a positive answer to a question of Kaimanovich. The equivalence is stated with respect to transverse invariant measures or some tangentially smooth measures. The latter include harmonic measures, and in this case the Folner condition has to be replaced by $\eta$-Folner (where the usual volume is modified by the modular form $\eta$ of the measure).

Sets satisfying the Central Sets Theorem

Central subsets of a discrete semigroup S have very strong combinatorial properties which are a consequence of the Central Sets Theorem . We investigate here the class of semigroups that have a subset with zero Folner density which satisfies the conclusion of the Central Sets Theorem. We show that this class includes any direct sum of countably many finite abelian groups as well as any subsemigroup of (ℝ,+) which contains ℤ. We also show that if S and T are in this class and either both are left cancellative or T has a left identity, then S×T is in this class. We also extend a theorem proved in (Beiglbock et al. in Topology Appl., to appear), which states that, if p is an idempotent in βℕ whose members have positive density, then every member of p satisfies the Central Sets Theorem. We show that this holds for all commutative semigroups. Finally, we provide a simple elementary proof of the fact that any commutative semigroup satisfies the Strong Folner Condition.

Group amenability properties for von Neumann algebras

In his study of amenable unitary representations, M.E.B. Bekka asked if there is an analogue for such representations of the remarkable fixed-point property for amenable groups. In this paper, we prove such a fixed-point theorem in the more general context of a G-amenable von Neumann algebra M, where G is a locally compact group acting on M. The Folner conditions of Connes and Bekka are extended to the case where M is semifinite and admits a faithful, semifinite, normal trace which is invariant under the action of G.

Folner Conditions, Nuclearity, and Subexponential Growth inC*-Algebras

Answering a first question of Voiculescu (On the existence of quasi-central approximate units relative to normed ideals,J. Funct. Anal.91(1990), 1–36) Vaillant 8?and Kirchberg (OnC*-algebras having subexponential, polynomial and linear growth,Invent. Math.108(1992), 635–652) showed that subexponential growth implies nuclearity in theC*-context. Voiculescu suggested a Følner type condition forC*-algebras and asked about the relation to growth and nuclearity. In this work we clarify the relation among subexponential growth phenomena, this Følner condition suggested by Voiculescu, weak filtrability in the sense of Arveson and Bedos, and nuclearity in theC*-algebra context.

Structural properties of inner amenable groups

For infinite discrete groups, Effros introduced the notion of inner amenability which gives a new classification of discrete groups. The inner amenability is a considerably weaker condition than amenability, but closely related to the quite deep property Γ of groups. In this paper the author investigates the structures of inner amenable groups by theoretical set theory. A sequence of characterizations of inner amenable groups is given here by developing the well-known Folner's conditions for amenable locally compact groups.

Entropy of stochastic processes homogeneous with respect to a commutative group of transformations

In order to define the entropy of a stochastic field homogeneous with respect to a countable commutative group of transformations G, one fixes a sequence {An} of finite subsets of the group G and considers the upper limit of the sequence of mean entropies of the iterates of the decomposition P. i.e.,\(\mathop {\lim }\limits_{n \to \infty } |A_n |{}^{ - 1}H \cdot (\mathop V\limits_{g \in A_n } TgP)\), where ¦An¦ is the number of elements in An. It is proved that for a fixed stochastic field and all sequences {An} satisfying the Folner condition, the limit of the means exists and is unique. If the sequence {An} is such that for all stochastic fields invariant under G, the entropy calculated in terms of it is the same as that calculated for a Folner-sequence, then {An} satisfies the Folner condition. In the case when G is a ν-dimensional lattice Zν, the Folner condidition coincides with the Van Hove condition.

Folner's Condition for Exponentially Bounded Groups.

Folner's Condition for Exponentially Bounded Groups.

A Generalization of Folner's Condition.

A Generalization of Folner's Condition.

Ratio properties in locally compact amenable groups

0. Introduction and summary. Let r be a locally compact (Hausdorff) topological group, and let 1*1 denote a fixed left Haar measure on the Borel sets of 1' (in the sense of [1, ?51]). It is known (see [2]) that the existence of a left invariant mean on LP(r) is equivalent to the Folner Condition (FC): To any E > 0 and any compact set Cc r, there corresponds a Borel set V of finite nonzero measure such that

Semigroups Satisfying a Strong Folner Condition

Semigroups Satisfying a Strong Folner Condition

A New Proof for Folner's Condition for Maximally Almost Periodic Groups.

A New Proof for Folner's Condition for Maximally Almost Periodic Groups.

Folner's Conditions for Amenable Semi-Groups.

Folner's Conditions for Amenable Semi-Groups.