Strong Arens irregularity of Beurling algebras with a locally convex topology

Let G be a locally compact group with a weight function ω. Recently, we have shown that the Banach space L₀^∞ (G,1/ω) can be identified with the strong dual of L¹(G, ω)equipped with some locally convex topologies τ. Here we use this duality to introduce an Arens multiplication on (L¹(G, ω), τ)**, and prove that the topological center of (L¹(G, ω), τ)** is (L¹(G, ω); this enables us to conclude that (L¹(G, ω), τ) is Arens regular if and only if G is discrete. We also give a characterization for Arens regularity of L₀^∞ (G, 1/ω)¹. Received: 8 March 2005     

On banach spaces which containl 1(τ) and types of measures on compact spaces

Two closely related results are presented, one of them concerned with the connection between topological and measure-theoretic properties of compact spaces, the other being a non-separable analogue of a result of Peŀczyński's about Banach spaces containingL ¹. Let τ be a regular cardinal satisfying the hypothesis that κ^ω<τ whenever κ<τ. The following are proved: 1) A compact spaceT carries a Radon measure which is homogeneous of type τ, if and only if there exists a continuous surjection ofT onto [0, 1]^τ. 2) A Banach spaceX has a subspace isomorphic tol ¹(τ) if and only ifX ^∗ has a subspace isomorphic toL ¹({0, 1}^τ). An example is given to show that a more recent result of Rosenthal's about Banach spaces containingl ¹ does not have an obvious transfinite analogue. A second example (answering a question of Rosenthal's) shows that there is a Banach spaceX which contains no copy ofl ¹ (ω₁), while the unit ball ofX ^∗ is not weakly^∗ sequentially compact.     

On Two Questions About Topological (and Nonstandard) Extensions

We construct, under MA, a non-Hausdorff (T₁-)topological extension ^*ω of ω, such that every function from ω to ω extends uniquely to a continuous function from ^*ω to ^*ω. We also show (in ZFC) that for every nontrivial topological extension ^*X of a countable set X there exists a topology τ_f on ^*X, strictly finer than the Star topology, and such that (^*X, τ_f) is still a topological extension of X with the same function extensions ^*f. This solves two questions raised by M. Di Nasso and M. Forti.     

On Two Questions About Topological (and Nonstandard) Extensions

We construct, under MA, a non-Hausdorff (T₁-)topological extension ^*ω of ω, such that every function from ω to ω extends uniquely to a continuous function from ^*ω to ^*ω. We also show (in ZFC) that for every nontrivial topological extension ^*X of a countable set X there exists a topology τ_f on ^*X, strictly finer than the Star topology, and such that (^*X, τ_f) is still a topological extension of X with the same function extensions ^*f. This solves two questions raised by M. Di Nasso and M. Forti.     

Uniformly complementedl p n ’s in quasi-reflexive Banach spaces

It is shown that for every non-reflexive Banach spaceX withX ^**/X reflexive there exists a uniformly bounded sequence of projections {P _n } _n=1 ^∞ whose ranges are uniformly isomorphic to {l _p ⁿ } _{n = 1} ^∞ either forp=1, orp=2 or forp=∞. The proof uses knowledge of the transfinite dualX ^ω, ESA Schauder decompositions and proof of a similar statement for spaces with an unconditional basis due to Tzafriri.     

Topological ergodic decompositions and applications to products of powers of a minimal transformation

For a minimal distal flow (X, T) and a positive integern, let be the largest distal factor of ordern. The existence of a denseG _δ subset ω ofX is shown, such that forx ∈ ω the orbit closure of (x,x,...,x) ∈ X ⁿ⁺¹ under τ =T ×T ² ... ×T ⁿ⁺¹ is π-saturated. In fact, an analogous statement for a general minimal flow is proved in terms of its PI-tower. On the way we get some topological “ergodic” decomposition theorems.     

The limiting distribution of the trace of a random plane partition

We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τ _n = τ _n (ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τ _n − c ₀ n ^2/3)/c ₁ n ^1/3 log^1/2 n converges weakly, as n → ∞, to the standard normal distribution, where c ₀ = ζ(2)/ [2ζ(3)]^2/3, c ₁ = √(1/3/) [2ζ(3)]^1/3 and ζ(s) = Σ _j=1^∞ j ^−s . Partial support given by the National Science Fund of the Bulgarian Ministry of Education and Science, grant No. VU-MI-105/2005.     

Iterated forcing in quadratic form theory

In [Sp1] and [B/Sp] it has been shown that the existence of quadratic spaces of uncountable dimension over finite or countable fields sharing the property that every infinite dimensional subspace has its orthogonal complement of at most countable dimension is independent of the axioms of ZFC set theory. Such a space will be called astrong Gross space in the sequel. Cardinal invariants of the continuum decide whether strong Gross spaces exist or not. Namely, when b=ω₁ a strong Gross space of dimension ℵ₁ exists. When p>ω₁ such spaces do not exist. Here we answer the question what happens with strong Gross spaces in case b>ω₁ or p=ω₁. This work forms part of the author’s Habilitationsschrift at the ETH Zürich. The author is supported by the Basic Research Foundation of the Israel Academy of Sciences.     

The weighted Herz-type Hardy spaces hK q α,p (ω1,ω2)

Let 1<q<∞, n(1−1/q)≤α<∞, 0<p<∞ and ω₁,ω₂ ɛA ₁(R ⁿ ) (the Muckenhoupt class). In this paper, the author introduce the weighted Herz-type Hardy spaces hk _q ^α,p (gw₁,ω₂) and present their atomic decomposition. Using the atomic decomposition, the author find out their dual spaces, establish the boundedness on these spaces of the pseudo-differential operators of order zero and show thatD(R ⁿ ), the class of C^∞(Rⁿ)-functions with compactly support, is dense inhK _q ^α,p (ω₁,ω₂) and there is a subsequence, which converges in distrbutional sense to some distribution ofhK _q ^α,p (ω₁,ω₂), of any bounded sequence inhK _q ^α,p (ω₁,ω₂). In addition, the author also set up the boundedness of some non-linear quantities in compensated compactness. Supported by the NECF and the NECF and the NNSF of China.     

On dualL 1-spaces and injective bidual Banach spaces

In a previous paper (Israel J. Math.28 (1977), 313–324), it was shown that for a certain class of cardinals τ,l ¹(τ) embeds in a Banach spaceX if and only ifL ¹([0, 1]^τ) embeds inX ^*. An extension (to a rather wider class of cardinals) of the basic lemma of that paper is here applied so as to yield an affirmative answer to a question posed by Rosenthal concerning dual ℒ₁-spaces. It is shown that ifZ ^* is a dual Banach space, isomorphic to a complemented subspace of anL ¹-space, and κ is the density character ofZ ^*, thenl ¹(κ) embeds inZ ^*. A corollary of this result is that every injective bidual Banach space is isomorphic tol ^∞(κ) for some κ. The second part of this article is devoted to an example, constructed using the continuum hypothesis, of a compact spaceS which carries a homogeneous measure of type ω₁, but which is such thatl ¹(ω₁) does not embed in ℰ(S). This shows that the main theorem of the already mentioned paper is not valid in the case τ = ω₁. The dual space ℰ(S)^* is isometric to , and is a member of a new isomorphism class of dualL ¹-spaces.     

Covering properties which,under weak diamond principles,constrain the extents of separable spaces

We show that separable, locally compact spaces with property (a) necessarily have countable extent — i.e., have no uncountable closed, discrete subspaces — if the effective weak diamond principle ⋄(ω,ω,<) holds. If the stronger, non-effective, diamond principle Φ(ω,ω,<) holds then separable, countably paracompact spaces also have countable extent. We also give a short proof that the latter principle implies there are no small dominating families in ^ω ₁ ω.     

The Helmholtz decomposition of weightedLq-spaces for Muckenhoupt weights

We consider weights of Muckenhoupt classA _q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝⁿ we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve the weak Neumann problem for the Laplace equation in weighted spaces on ℝⁿ, ℝⁿ ₊, on bounded and on exterior domains Ω with boundary of classC ¹, which will yield the Helmholtz decomposition ofL _ω ^q(Ω)ⁿ for general ω∈A _q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of Farwig and Sohr [2] where the Helmholtz decomposition ofL _ω ^p(Ω)ⁿ is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood of ϖΩ.

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Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA _q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝⁿ, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝⁿ, ℝⁿ ₊ in domini limitati ed in domini esterni con frontiera di classeC ¹, che conduce alla decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ per un qualsiasi ω∈A _q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.

     

Absolutely continuous invariant measures for piecewise expandingC 2 transformations inR N

LetS be a bounded region inR ^N and let ℊ={S _i} _i ^=1/m be a partition ofS into a finite number of subsets having piecewiseC ² boundaries. We assume that whereC ² segments of the boundaries meet, the angle subtended by tangents to these segments at the point of contact is bounded away from 0. Letτ:S →S be piecewiseC ² on ℊ and expanding in the sense that there exists 0<σ< 1 such that for anyi=1, 2, ...,m, ‖Dτ _i ⁻¹ ‖<σ, whereDτ _i ⁻¹ is the derivative matrix ofτ _i ⁻¹ and ‖ ‖ is the euclidean matrix norm. The main result provides an upper bound onσ which guarantees the existence of an absolutely continuous invariant measure forτ. The research of the second author was supported by NSERC and FCAR grants.     

Commutator equations in free groups

Letf ₁, …,f _n be free generators of a free groupF. We consider the equation [z ₁, …,z _n]_ω. where ω and ω′ indicate the disposition of brackets in the higher commutators [z ₁, …,z _n]_ω and [f ₁, …,f _n]_ω. We give a necessary and sufficient condition on ω and ω′ for the existence of solutions of this equation. It is also shown that for any solutionz ₁=r₁, …,z _z=r _n we have <r ₁, …,r _n>=〈f ₁, …f _n〉.     

Instances of the Conjecture of Chang

We consider various forms of the Conjecture of Chang. Part A constitutes an introduction. Donder and Koepke have shown that if ρ is a cardinal such that ρ ≧ ω₁, and (ρ⁺⁺,ρ⁺↠(ρ⁺, ρ), then 0⁺ exists. We obtain the same conclusion in Part B starting from some other forms of the transfer hypothesis. As typical corollaries, we get: Theorem A.Assume that there exists cardinals λ, κ, such that λ ≧ K ⁺ ≧ω₂ and (λ⁺, λ)↠(K ⁺,K. Then 0⁺ exists. Theorem B.Assume that there exists a singularcardinal κ such that(K ⁺,K↠(ω₁, ω₀. Then 0⁺ exists. Theorem C.Assume that (λ ⁺⁺, λ). Then 0⁺ exists (also ifK=ω ₀. Remark. Here, as in the paper of Donder and Koepke, “O⁺ exists” is a matter of saying that the hypothesis is strictly stronger than “L(μ) exists”. Of course, the same proof could give a few more sharps overL(μ), but the interest is in expecting more cardinals, coming from a larger core model. Theorem D.Assume that (λ ⁺⁺, λ)↠(K ⁺, K) and thatK≧ω ₁. Then 0⁺ exists. Remark 2. Theorem B is, as is well-known, false if the hypothesis “κ is singular” is removed, even if we assume thatK≧ω ₂, or that κ is inaccessible. We shall recall this in due place. Comments. Theorem B and Remark 2 suggest we seek the consistency of the hypothesis of the form:K ⁺, K↠(ω_n +1, ω_n), for κ singular andn≧0. 0266 0152 V 3 The consistency of several statements of this sort—a prototype of which is (N _ω+1,N _ω)↠(ω₁, ω₀) —have been established, starting with an hypothesis slightly stronger than: “there exists a huge cardinal”, but much weaker than: “there exists a 2-huge cardinal”. These results will be published in a joint paper by M. Magidor, S. Shelah, and the author of the present paper.     

Triangular finite elements of HCT type and classC ρ

Let τ be some triangulation of a planar polygonal domain Ω. Given a smooth functionu, we construct piecewise polynomial functionsv∈C ^ρ(Ω) of degreen=3 ρ for ρ odd, andn=3ρ+1 for ρ even on a subtriangulation τ₃ of τ. The latter is obtained by subdividing eachT∈ρ into three triangles, andv/T is a composite triangular finite element, generalizing the classicalC ¹ cubic Hsieh-Clough-Tocher (HCT) triangular scheme. The functionv interpolates the derivatives ofu up to order ρ at the vertices of τ. Polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements of this type.     

Random ergodic theorems and real cocycles

We study mean convergence of ergodic averages associated to a measure-preserving transformation or flow τ along the random sequence of times κ _n (ω) given by the Birkhoff sums of a measurable functionF for an ergodic measure-preserving transformationT. We prove that the sequence (k _n(ω)) is almost surely universally good for the mean ergodic theorem, i.e., that, for almost every, ω, the averages (*) converge for every choice of τ, if and only if the “cocycle”F satisfies a cohomological condition, equivalent to saying that the eigenvalue group of the “associated flow” ofF is countable. We show that this condition holds in many natural situations. When no assumption is made onF, the random sequence (k _n(ω)) is almost surely universally good for the mean ergodic theorem on the class of mildly mixing transformations τ. However, for any aperiodic transformationT, we are able to construct an integrable functionF for which the sequence (k _n(ω)) is not almost surely universally good for the class of weakly mixing transformations.     

Stopping rules forx n/n and related problems

The following results illustrate the problems with which this note deals. Letx _n (n=1, 2, ...) be non-negative, independent, identically distributed random variables, letβ>1 andEx ₁ ^β <∞. Then there exists a stopping ruleτ withP{τ<∞}=1, which maximizesE x _t/t among all stopping rulest. Moreover, the same rule maximizesE max (x ₁, ...,x _t)/t andE max (x ₁,..,x _τ)/τ=Ex _τ/τ Research supported in part by Grant GP-5705 of the National Science Foundation, USA.     

On the monadic theory ofω 1 without A.C.

Buchi inLecture Notes in Mathematics, Decidable Theories II (1973) by using A.C. characterized the theoriesMT[β, <] forβ<ω ₁ and showed thatMT[ω ₁, <] is decidable. We extend Buchi’s results to a larger class of models of ZF (without A.C.) by proving the following under ZF only: (1) There is a choice function which chooses a “good” run of an automaton on countable input (Lemma 5.1). It follows that Buchi’s results cocerning countable ordinals are provable within ZF. (2) Let U.D. be the assertion that there exists a uniform denumeration ofω ₁ (i.e. a functionf: ω ₁ → ω ₁ ^ω such that for everyα<ω ₁,f(α) is a function fromω ontoα). We show that U.D. can be stated as a monadic sentence, and thereforeω ₁ is characterizable by a sentence. (3) LetF be the filter of the cofinal closed subsets ofω ₁. We show that if U.D. holds thenMT[ω ₁, <] is recursive in the first order theory of the boolean algebraP (ω ₁)/F. (We can effectively translate each monadic sentence Σ to a boolean sentenceσ such that [ω ₁, <] ⊨ Σ iffP(ω ₁)/F⊨σ). (4) As every complete boolean algebra theory is recursive we have that in every model of ZF+U.D.,MT[ω ₁, <] is recursive. All our proofs are within ZF. Buchi’s work is often referred to. Following Buchi, the main tool is finite automata. We don’t deal withMT[ω ₁, <] forω ₁ which doesn’t satisfy U.D. The results in this paper appeared in the author’s M.Sc. thesis, which was prepared at the Hebrew University under the supervision of Professor M. Rabin.     

Universal lattices and property tau

We prove that the universal lattices – the groups G=SL_d(R) where R=ℤ[x₁,...,x_k], have property τ for d≥3. This provides the first example of linear groups with τ which do not come from arithmetic groups. We also give a lower bound for the τ-constant with respect to the natural generating set of G. Our methods are based on bounded elementary generation of the finite congruence images of G, a generalization of a result by Dennis and Stein on K₂ of some finite commutative rings and a relative property T of . Mathematics Subject Classification (2000) 20F69, 13M05, 19C20, 20G05, 20H05