A poisson formula for solvable Lie groups

Given a probability measure μ on a locally compact second countable groupG the space of bounded μ-harmonic functions can be identified withL ^∞(η, α) where (η, α) is a BorelG-space with a σ-finite quasiinvariant measure α. Our goal is to show that when μ is an arbitrary spread out probability measure on a connected solvable Lie groupG then the μ-boundary (η, α) is a contractive homogeneous space ofG. Our approach is based on a study of a class of strongly approximately transitive (SAT) actions ofG. A BorelG-space η with a σ-finite quasiinvariant measure α is called SAT if it admits a probability measurev≪α, such that for every Borel set A with α(A)≠0 and every ε>0 there existsg∈G with ν(gA)>1−ε. Every μ-boundary is a standard SATG-space. We show that for a connected solvable Lie group every standard SATG-space is transitive, characterize subgroupsH⊆G such that the homogeneous spaceG/H is SAT, and establish that the following conditions are equivalent forG/H: (a)G/H is SAT; (b)G/H is contractive; (c)G/H is an equivariant image of a μ-boundary.     

Group measure space decomposition of II1 factors and W*-superrigidity

We prove a “unique crossed product decomposition” result for group measure space II₁ factors L ^∞(X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family G\mathcal{G}, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T _n denotes the group of upper triangular matrices in PSL (n,ℤ), then any free, mixing p.m.p. action of G = \operatornamePSL(n,\mathbbZ)*_Tn\operatornamePSL(n,\mathbbZ)\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z}) is W^∗-superrigid, i.e. any isomorphism between L ^∞(X)⋊Γ and an arbitrary group measure space factor L ^∞(Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family G\mathcal{G}, the Bernoulli actions of Γ are W^∗-superrigid.     

Balleans of topological groups

A subset S of a topological group G is called bounded if, for every neighborhood U of the identity of G, there exists a finite subset F such that S ⊆ FU, S ⊆ UF. The family of all bounded subsets of G determines two structures on G, namely the left and right balleans B _l (G) and B _r (G), which are counterparts of the left and right uniformities of G. We study the relationships between the uniform and ballean structures on G, describe all topological groups admitting a metric compatible both with uniform and ballean structures, and construct a group analogue of Higson’s compactification of a proper metric space.     

On some special measures on βG

LetG be a discrete abelian group,Ĝ the character group ofG, andl ^∞(G)^* the conjugate ofl ^∞(G) equipped with an Arens product. In many cases, we can find unitary functionsf such that χf is almost convergent to zero for all χ∈Ĝ. Some of these functions are then used to produce elements μ∈l ^∞(G)^* such that γμ=0 whenever γ is an annihilator ofC ₀(G). Regarded as Borel measures on βG, these elements satisfyxμ=0 for allx∈βG/G. They belong to the radical ofl ^∞(G)^*, and each of them generates a left ideal ofl ^∞(G)^* that contains no minimal left ideal.     

The asymptotic σ-algebra of a recurrent random walk on a locally compact group

Let μ be a probability measure on a locally compact second countable groupG defining a recurrent (but not necessarily Harris) random walk. Denote byG ^∞ the space of paths and byB ^(a)the asymptotic σ-algebra. Let the starting measure be equivalent to the Haar measure and writeQ for the corresponding Markov measure onG ^∞. We prove thatL ^∞(G^∞, B^(a), Q) is in a canonical way isomorphic toL ^∞(G/N) whereN is the smallest closed normal subgroup ofG such that μ(zN)=1 for somez∈G. The groupG/N is either a finite cyclic group with generatorzN or a compact abelian group having the cyclic group as a dense subgroup. As a corollary we obtain that the set of all φ∈L ¹(G) such that coincides with the kernel of the canonical mapping ofL ¹(G)ontoL ¹(G/N). In particular, when μ is aperiodic, i.e.,G=N, then the random walk is mixing: for every φ∈L ¹(G) with ∝ φ=0.     

Possible limit laws for entrance times of an ergodic aperiodic dynamical system

LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ ₀ ^∞ G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ _U (x)=inf{k⩾1:T ^k xεU}, and defineG _U (t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U _n ) _n≥1 of neighbourhoods ofx such that {x}=∩ _n U _n , and for anyG ∈G, there exists a subsequence (n _k ) _k≥1 withG _{U n k} ↑U weakly. We also construct a uniquely ergodic Toeplitz flowO(x ^∞,S, μ), the orbit closure of a Toeplitz sequencex ^∞, such that the above conclusion still holds, with moreover the requirement that eachU _n be a cylinder set. In memory of Anzelm Iwanik     

Time Regularity for Random Walks on Locally Compact Groups

Let G be a compactly generated, locally compact group, and let T be the operator of convolution with a probability measure μ on G. Our main results give sufficient conditions on μ for the operator T to be analytic in L ^p (G), 1 < p < ∞, where analyticity means that one has an estimate of form for all n = 1, 2, ... in L ^p operator norm. Counterexamples show that analyticity may not hold if some of the conditions are not satisfied.     

Distal actions and shifted convolution property

A locally compact group G is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure μ on G, either sup _x∈G μ ⁿ (Cx) → 0 for all compact subsets C of G, or there exist x ∈ G and a compact subgroup K normalised by x such that μ ⁿ x ⁻ⁿ → ωK, the normalised Haar measure on K. We first consider distality of factor actions of distal actions. It is shown that this holds in particular for factors by compact groups invariant under the action and for factors by the connected component of the identity. We then characterize groups having SCP in terms of a readily verifiable condition on the conjugation action (pointwise distality). This gives some interesting corollaries to distality of certain actions and Choquet-Deny measures which actually motivated SCP and pointwise distal groups. We also relate distality of actions on groups to that of the extensions on the space of probability measures.     

Characterization of Operators on the Dual of Hypergroups which Commute with Translations and Convolutions

For a locally compact group G, L^1 (G) is its group algebra and L^∞(G) is the dual of L^1 (G).Lau has studied the bounded linear operators T:L^∞(G)→L^∞(G) which commute with convolutions and translations. For a subspace H of L^∞(G), we know that M(L^∞(G),H), the Banach algebra of all bounded linear operators on L^∞(G) into H which commute with convolutions, has been studied by Pyre and Lau. In this paper, we generalize these problems to L(K)^*, the dual of a hypergroup algebra L(K) in a very general setting, i.e. we do not assume that K admits a Haar measure. It should be noted that these algebras include not only the group algebra L^1(G) but also most of the semigroup algebras.Compact hypergroups have a Haar measure, however, in general it is not known that every hypergroup has a Haar measure. The lack of the Haar measure and involution presents many difficulties; however,we succeed in getting some interesting results.     

Random walks on almost connected locally compact groups: Boundary and convergence

We prove that, given an arbitrary spread out probability measure μ on an almost connected locally compact second countable groupG, there exists a homogeneous spaceG/H, called the μ-boundary, such that the space of bounded μ-harmonic functions can be identified withL ^∞ (G/H). The μ-boundary is an amenable contractive homogeneous space. We also establish that the canonical projection onto the μ-boundary of the right random walk of law μ always converges in probability and, whenG is amenable, it converges almost surely. The μ-boundary can be characterised as the largest homogeneous space among those homogeneous spaces in which the canonical projection of the random walk converges in probability.     

On spectral characterizations of amenability

We show that a measuredG-space (X, μ), whereG is a locally compact group, is amenable in the sense of Zimmer if and only if the following two conditions are satisfied: the associated unitary representationπ _X ofG intoL ²(X, μ) is weakly contained into the regular representationλ _G and there exists aG-equivariant norm one projection fromL∞(X×X) ontoL∞(X). We give examples of ergodic discrete group actions which are not amenable, althoughπ _X is weakly contained intoλ _G.     

Proprietes generiques de processus croises

LetS _φ be the skew product transformation(x, g)↦(Sx, gφ(x)) defined on Ω×G, where Ω is a compact metric space,G a compact metric group with its Haar measureh. IfS is a μ-continuous transformation where μ is a Borel measure on Ω, ergodic with respect toS, we study the setE ₀ of μ-continuous applications φ:Ω→G such that μ⩀h is ergodic (with respect toS _φ). For example,E ₀ is residual in the group of μ-continuous applications from Ω toG with the uniform convergence topology. We also study the weakly mixing case. Some arithmetic applications are given.     

Lacunary Fourier series and a qualitative uncertainty principle for compact Lie groups

We define lacunary Fourier series on a compact connected semisimple Lie group G. If f ∈ L ¹(G) has lacunary Fourier series and f vanishes on a non empty open subset of G, then we prove that f vanishes identically. This result can be viewed as a qualitative uncertainty principle.     

Probability measures on [SIN] groups and some related ideals in group algebras

Given a locally compact group G, let J(G){\cal J}(G) denote the set of closed left ideals in L ¹(G), of the form J _μ = [L¹(G) * (δ _e − μ)]⁻, where μ is a probability measure on G. Let J_d(G)={\cal J}_d(G)= {J_m;m is discrete}\{J_{\mu};\mu\ {\rm is discrete}\} , J_a(G)={J_m;m is absolutely continuous}{\cal J}_a(G)=\{J_{\mu};\mu\ {\rm is absolutely continuous}\} . When G is a second countable [SIN] group, we prove that J(G)=J_d(G){\cal J}(G)={\cal J}_d(G) and that J_a(G){\cal J}_a(G) , being a proper subset of J(G){\cal J}(G) when G is nondiscrete, contains every maximal element of J(G){\cal J}(G) . Some results concerning the ideals J _μ in general locally compact second countable groups are also obtained.     

Extreme Jensen measures

Let Ω be an open subset of R ^d , d≥2, and let x∈Ω. A Jensen measure for x on Ω is a Borel probability measure μ, supported on a compact subset of Ω, such that ∫u  dμ≤u(x) for every superharmonic function u on Ω. Denote by J _x (Ω) the family of Jensen measures for x on Ω. We present two characterizations of ext(J _x (Ω)), the set of extreme elements of J _x (Ω). The first is in terms of finely harmonic measures, and the second as limits of harmonic measures on decreasing sequences of domains. This allows us to relax the local boundedness condition in a previous result of B. Cole and T. Ransford, Jensen measures and harmonic measures, J. Reine Angew. Math. 541 (2001), 29–53. As an application, we give an improvement of a result by Khabibullin on the question of whether, given a complex sequence {α _n } _n=1 ^∞ and a continuous function , there exists an entire function f≢0 satisfying f(α _n )=0 for all n, and |f(z)|≤M(z) for all z∈C.     

Joint ergodicity and mixing

The study of jointly ergodic measure preserving transformations of probability spaces, begun in [1], is continued, and notions of joint weak and strong mixing are introduced. Various properties of ergodic and mixing transformations are shown to admit analogues for several transformations. The case of endomorphisms of compact abelian groups is particularly emphasized. The main result is that, given such commuting endomorphisms σ₁σ₂,...,σ, ofG, the sequence ((1/N)Σ _n=0 ^N−1 σ ₁ ⁿ f ¹·σ ₂ ⁿ f ²· ··· · σ _s ⁿ f ^sconverges inL ²(G) for everyf ₁,f ₂,…,f _s∈L ^∞(G). If, moreover, the endomorphisms are jointly ergodic, i.e., if the limit of any sequence as above is Π _i=1 ^s ∫ _G f ₁ d μ, where μ is the Haar measure, then the convergence holds also μ-a.e.     

Uniformly converging random variables for weakly converging laws

 Let {P _n , n ?ℕ} be a sequence of Borel probability measures on a compact and connected metric space X. We show that in case the measures P _n converge weakly to a fully supported limit measure P, there exist uniformly converging random variables X _n , n ?ℕ with these given laws. Connectivity and compactness are necessary conditions for our theorem to hold. We also present a decent generalization. We prove our theorem by means of a comparison of the Prokhorov and the so-called minimal L ^∞ metric. Then we only need to use the Strassen-Dudley theorem and Kellerer's measure extension theorem for decomposable families. Received: 2 November 2000 / Revised version: 5 January 2002/ Published online: 1 July 2002     

A Spectral Gap Property for Random Walks Under Unitary Representations

Let G be a locally compact group and μ a probability measure on G, which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $\pi ,\mathcal{H}Let G be a locally compact group and μ a probability measure on G, which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation p,H\pi ,\mathcal{H} of G, we study spectral properties of the operator π(μ) acting on H\mathcal{H} Assume that μ is adapted and that the trivial representation 1 _G is not weakly contained in the tensor product p?[`(p)]\pi\otimes \overline\pi We show that π(μ) has a spectral gap, that is, for the spectral radius r_spec(p(m))r_{\rm spec}(\pi(\mu)) of π(μ), we have r_spec(p(m)) < 1.r_{\rm spec}(\pi(\mu))< 1. This provides a common generalization of several previously known results. Another consequence is that, if G has Kazhdan’s Property (T), then r_spec(p(m)) < 1r_{\rm spec}(\pi(\mu))< 1 for every unitary representation π of G without finite dimensional subrepresentations. Moreover, we give new examples of so-called identity excluding groups.     

A simple map with no prime factors

An ergodic measure-preserving transformationT of a probability space is said to be simple (of order 2) if every ergodic joining λ ofT with itself is eitherμ×μ or an off-diagonal measureμ _S , i.e.,μ _S (A×B)=μ(A∩S ^;−n ;B) for some invertible, measure preservingS commuting withT. Veech proved that ifT is simple thenT is a group extension of any of its non-trivial factors. Here we construct an example of a weakly mixing simpleT which has no prime factors. This is achieved by constructing an action of the countable Abelian group ℤ⊕G, whereG=⊕ _i=1 ^∞ ℤ₂, such that the ℤ-subaction is simple and has centralizer coinciding with the full ℤ⊕G-action.     

Courants invariants par une action propre

The purpose of this paper is to characterise the invariant sections-distributions by a proper action. More precisely, we show that if G is a connected Lie group acting on a differentiable vector bundle E→V such that the induced action on V is proper, then the topological vector space of the G-invariant linear functionals (on the space of C ^∞ sections with compact support) equipped with the induced weak-topology (resp. the strong-topology), is isomorphic to the weak (resp. strong) topological dual of the space (of all G-invariant sections σ with compact quotient supp(σ)/G) equipped with a suitable topology; this coincides with the usual C ^∞-topology if the orbit space is compact, and with the Schwartz-topology if the group G is compact. Received: 8 June 1998 / Revised version: 22 September 1998