The Derivation Problem for Group Algebras of Connected Locally Compact Groups

The derivation problem for a locally compact group G is to decidewhether for each derivation D from L¹(G) into L¹(G) there isa bounded measure µM(G) with D(a) = aµ–µa(a L¹(G)). In this paper we obtain an affirmative answer forthe case of connected groups. To explain the contents of thispaper we give an equivalent formulation of the problem. Supposethat the group G acts as a group of homeomorphisms of the locallycompact space X. Related to this there is an action of G onM(X). A bounded crossed homomorphism from G to M(X) is a map with bounded range and satisfying (gh) = g(h)+(g) (g, h G).The problem for bounded crossed homomorphisms is to decide iffor each such there is an element µ of M(X) with (g)= gµ– µ (g G). The derivation problem isequivalent to this bounded crossed homomorphism problem forthe special case X = G where G acts on X by conjugation (togetherwith some mild continuity hypotheses about the map :GM(X) whichare often automatically satisfied). The bounded crossed homomorphismproblem always has a positive solution if G is amenable anda closely related calculation shows that in solving the boundedcrossed homomorphism problem we need only solve it for functions which are zero on H where H is a given amenable subgroup ofG. It can happen that this condition of being zero on H forces to be zero even when H is a comparatively small subgroup ofG. If h is an element of G such that ‘hⁿx ’ asn for all x X then for any two measures µ and , forlarge values of n, µ and hⁿ have little overlap so ||µ+ hⁿ|| ||µ|| + ||||. Thus if H is the subgroup generatedby h, for any g G .     

Constructible Functions on Artin Stacks

Let be an algebraically closed field, let X be a -variety,and let X() be the set of closed points in X. A constructibleset C in X() is a finite union of subsets Y() for subvarietiesY in X. A constructible function f : X() has f(X()) finiteand f^–1(c) constructible for all c 0. Write CF(X) forthe vector space of such f. Let : X Y and : Y Z be morphismsof -varieties. MacPherson defined a linear pushforward CF(): CF(X) CF(Y) by ‘integration’ with respect tothe topological Euler characteristic. It is functorial, thatis, CF( ) = CF() CF(). This was extended to of characteristiczero by Kennedy. This paper generalizes these results to -schemes and Artin -stackswith affine stabilizer groups. We define the notions of Eulercharacteristic for constructible sets in -schemes and -stacks,and pushforwards and pullbacks of constructible functions, withfunctorial behaviour. Pushforwards and pullbacks commute inCartesian squares. We also define pseudomorphisms, a generalizationof morphisms well suited to constructible functions problems.     

Evasion and Prediction II

A subgroup G Z exhibits the Specker phenomenon if every homomorphismG Z maps almost all unit vectors to 0. We give several combinatorialcharacterizations of the cardinal e, the size of the smallestG Z exhibiting the Specker phenomenon. We also prove the consistencyof > e, where is the unbounding number and e the evasionnumber. Our results answer several questions addressed by Blass.     

On Borel Sets in Function Spaces with the Weak Topology

It is proved that the duality map ,:(, weak)x(()*, weak*)R isnot Borel. More generally, the evaluation e:(C)(K),x KR, e(f,x) = f(x), is not Borel for any function space C(K) on a compactF-space. It is also shown that a non-coincidence of norm-Boreland weak-Borel sets in a function space does not imply thatthe duality map is non-Borel.     

Quadratic Integrals and Factorization of Linear Operators

We consider operators on the matrix-valued disc algebra. Weshow that any bounded linear operator to a Banach space X of cotype 2 induces a boundedoperator GX defined on some weighted Bergman space G on theunit disc. We give sufficient conditions on the weight forthe formal inclusion to be 2–C*-summing. Decompositions of T with respect to operator-valuedmeasures are obtained.     

Metrical Theorems on Prime Values of the Integer Parts of Real Sequences II

Let [ ] denote the integer part. Among other results in [3]we gave a complete solution to the following problem. PROBLEM. Given an increasing sequence a_n R⁺, n = 1, 2, ...,where a_n as n , are there infinitely many primes in the sequence[a_n] for almost all ?     

Regularity Conditions and Bernoulli Properties of Equilibrium States and g-Measures

When T : X X is a one-sided topologically mixing subshift offinite type and : X R is a continuous function, one can definethe Ruelle operator L : C(X) C(X) on the space C(X) of real-valuedcontinuous functions on X. The dual operator always has a probability measure as an eigenvectorcorresponding to a positive eigenvalue ( = with > 0). Necessary and sufficient conditionson such an eigenmeasure are obtained for to belong to twoimportant spaces of functions, W(X, T) and Bow (X, T). For example, Bow(X, T) if and only if is a measure with a certain approximateproduct structure. This is used to apply results of Bradleyto show that the natural extension of the unique equilibriumstate µ of Bow(X, T) has the weak Bernoulli propertyand hence is measure-theoretically isomorphic to a Bernoullishift. It is also shown that the unique equilibrium state ofa two-sided Bowen function has the weak Bernoulli property.The characterizations mentioned above are used in the case ofg-measures to obtain results on the ‘reverse’ ofa g-measure.     

Relative Completions and the Cohomology of Linear Groups Over Local Rings

For a discrete group G there are two well known completions.The first is the Malcev (or unipotent) completion. This is aprounipotent group U, defined over Q, together with a homomorphism : G U that is universal among maps from G into prounipotentQ-groups. To construct U, it suffices for us to consider thecase where G is nilpotent; the general case is handled by takingthe inverse limit of the Malcev completions of the G/^rG, where^•G denotes the lower central series of G. If G is abelian,then U = G Q. We review this construction in Section 2.     

Tensor Products and Operators in Spaces of Analytic Functions

Let X be an infinite dimensional Banach space. The paper provesthe non-coincidence of the vector-valued Hardy space H^p(T, X)with neither the projective nor the injective tensor productof H^p(T) and X, for 1 < p < . The same result is provedfor some other subspaces of L^p. A characterization is givenof when every approximable operator from X into a Banach spaceof measurable functions F(S) is representable by a functionF:S X* as x F(·), x. As a consequence the existenceis proved of compact operators from X into H^p(T) (1 p <) which are not representable. An analytic Pettis integrablefunction F:T X is constructed whose Poisson integral does notconverge pointwise.     

The Isolated Points of G{rho} and the w*-Strongly Exposed Points of P{rho}(G)0

Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C^*-algebraC^*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w^*-strongly exposed point of P(G)₀, where P(G)₀={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w^*-strongly exposed point of P(G)₀, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w^*-strongly exposed point of P(G)₀. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w^*-stronglyexposed point of P(G)₀, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F₂ on two generators), there isno closed point in its reduced dual space .     

Limits Along Parallel Lines and the Classical Fine Topology

The fine topology on Rⁿ (n2) is the coarsest topology for whichall superharmonic functions on Rⁿ are continuous. We refer toDoob [11, 1.XI] for its basic properties and its relationshipto the notion of thinness. This paper presents several theoremsrelating the fine topology to limits of functions along parallellines. (Results of this nature for the minimal fine topologyhave been given by Doob – see [10, Theorem 3.1] or [11,1.XII.23] – and the second author [15].) In particular,we will establish improvements and generalizations of resultsof Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihland Seidel [6], Schneider [21], Berman [7], and Armitage andNelson [4], and will also solve a problem posed by the latterauthors. An early version of our first result is due to Evans [12, p.234], who proved that, if u is a superharmonic function on R³,then there is a set ER²x{0}, of two-dimensional measure 0, suchthat u(x, y,·) is continuous on R whenever (x, y, 0)E.We denote a typical point of Rⁿ by X=(X' x), where X'R^n–1and xR. Let :RⁿR^n–1x{0} denote the projection map givenby (X', x) = (X', 0). For any function f:Rⁿ[–, +] andpoint X we define the vertical and fine cluster sets of f atX respectively by C_V(f;X)={l[–, +]: there is a sequence (t_m) of numbersin R\{x} such that t_mx and f(X', t_m)l}| and C_F(f;X)={l[–, +]: for each neighbourhood N of l in [–,+], the set f^–1(N) is non-thin at X}. Sets which are open in the fine topology will be called finelyopen, and functions which are continuous with respect to thefine topology will be called finely continuous. Corollary 1(ii)below is an improvement of Evans' result.     

An Access Theorem for Continuous Functions

Let f be a continuous function on an open subset of R² suchthat for every x there exists a continuous map : [–1,1] with (0) = x and f increasing on [–1, 1]. Thenfor every there exists a continuous map : [0, 1) suchthat (0) = y, f is increasing on [0; 1), and for every compactsubset K of , max{t : (t) K} < 1. This result gives an answerto a question posed by M. Ortel. Furthermore, an example showsthat this result is not valid in higher dimensions.     

The Bestvina-Brady Construction Revisited: Geometric Computation of {sum}-Invariants for Right-Angled Artin Groups

The starting point of our investigation is the remarkable paper[2] in which Bestvina and Brady gave an example of an infinitelyrelated group of type FP₂. The result about right-angled Artingroups behind their example is best interpreted by means ofthe Bieri–Strebel–Neumann–Renz -invariants. For a group G the invariants ⁿ(G) and ⁿ(G, Z) are sets of non-trivialhomomorphisms :GR. They contain full information about finitenessproperties of subgroups of G with abelian factor groups. Themain result of [2] determines for the canonical homomorphism, taking each generator of the right-angled Artin group G to1, the maximal n with ⁿ(G), respectively ⁿ(G, Z). In [6] Meier, Meinert and VanWyk completed the picture by computingthe full -invariants of right-angled Artin groups using as wellthe result of Bestvina and Brady as algebraic techniques from-theory. Here we offer a new account of their result which istotally geometric. In fact, we return to the Bestvina–Bradyconstruction and simplify their argument considerably by bringinga more general notion of links into play. At the end of thefirst section we re-prove their main result. By re-computingthe full -invariants, we show in the second section that thesimplification even adds some power to the method. The criterionwe give provides new insight on the geometric nature of the‘n-domination’ condition employed in [6].     

Majorations Uniformes de Normes D'Inverses Dans Les Algebres de Beurling

The Beurling algebras l¹(D,)(D=N,Z) that are semi-simple, withcompact Gelfand transform, are considered. The paper gives anecessary and sufficient condition (on ) such that l¹(D,) possessesa uniform quantitative version of Wiener's theorem in the sensethat there exists a function :]0,+[]0,+ such that, for everyinvertible element x in the unit ball of l¹(D,), we have ||x^–1||(r(x^–1)) r(x^–1) is the spectral radiusof x^–1.     

Contractions et Hyperdistributions a Spectre de Carleson

Let =(_n)_n1 be a log concave sequence such that lim inf_n+n/n^c>0for some c>0 and ((log _n)/n)_n1 is nonincreasing for some<1/2. We show that, if T is a contraction on the Hilbertspace with spectrum a Carleson set, and if ||T^–n||=O(_n)as n tends to + with _n11/(n log _n)=+, then T is unitary. Onthe other hand, if _n11/(n log _n)<+, then there exists a (non-unitary)contraction T on the Hilbert space such that the spectrum ofT is a Carleson set, ||T^–n||=O(_n) as n tends to +, andlim sup_n+||T^–n||=+.     

On the Boundedness and Compactness of a Class of Integral Operators

Let > 0. The operator of the form is considered, where the real weight function v(x) is locallyintegrable on R⁺ := (0, ). In case v(x) = 1 the operator coincideswith the Riemann–Liouville fractional integral, L^p L^qestimates of which with power weights are well known. This workgives L^p L^qboundedness and compactness criteria for the operatorT in the case 0 < p, q < , p > max(1/, 1).     

A Strong Law for the Largest Nearest-Neighbour Link between Random Points

Suppose that X₁, X₂, X₃, ... are independent random points inR^d with common density f, having compact support with smoothboundary , with f| continuous. Let Rⁿ_{i, k} denote the distancefrom X_i to its kth nearest neighbour amongst the first n points,and let M_{n, k} = max_in Rⁿ_{i, k}. Let denote the volume of theunit ball. Then as n , , almost surely If instead the points lie in a compact smooth d-dimensionalRiemannian manifold K, then nM^d_{n, k}/log n (min_Kf)^–1,almost surely.     

Herman Rings and Arnold Disks

For (,a) C* x C, let f_,a be the rational map defined by f_,a(z)= z² (az+1)/(z+a). If R/Z is a Brjuno number, we let D bethe set of parameters (,a) such that f_,a has a fixed Hermanring with rotation number (we consider that (e²ⁱ,0) D). Resultsobtained by McMullen and Sullivan imply that, for any g D, theconnected component of D(C* x (C/{0,1})) that contains g isisomorphic to a punctured disk. We show that there is a holomorphic injection F:DD such thatF(0) = (e²ⁱ ,0) and , where r is the conformal radius at 0 of the Siegel disk of the quadraticpolynomial z e²ⁱ z(1+z). As a consequence, we show that for a (0,1/3), if f_l,a has afixed Herman ring with rotation number and if m_a is the modulusof the Herman ring, then, as a0, we have e m_a=(r/a) + O(a). We finally explain how to adapt the results to the complex standardfamily z e^(a/2)(z-1/z).     

2-Ruled Calibrated 4-Folds in R7 and R8

This article introduces the notion of 2-ruled 4-folds: submanifoldsof Rⁿ fibred over a 2-fold by affine 2-planes. This is motivatedby a paper by Joyce and previous work of the present author.A 2-ruled 4-fold M is r-framed if an oriented basis is smoothlyassigned to each fibre, and then we may write M in terms oforthogonal smooth maps ₁,₂ : S^n–1 and a smooth map : Rⁿ. We focus on 2-ruled Cayley 4-folds in R⁸ as certainother calibrated 4-folds in R⁷ and R⁸ can be considered as specialcases. The main result characterizes non-planar, r-framed, 2-ruledCayley 4-folds, using a coupled system of nonlinear, first-order,partial differential equations that ₁ and ₂ satisfy, and anothersuch equation on which is linear in . We give a means of constructing2-ruled Cayley 4-folds starting from particular 2-ruled Cayleycones using holomorphic vector fields. This is used to giveexplicit examples of U(1)-invariant 2-ruled Cayley 4-folds asymptoticto a U(1)³-invariant 2-ruled Cayley cone. Examples are alsogiven based on ruled calibrated 3-folds in C³ and R⁷ and complexcones in C⁴.     

An egg-yolk principle and exponential integrability for quasiregular mappings

Quasiregular mappings f:ⁿⁿ are a natural generalization of analyticfunctions from complex analysis and provide a theory which isrich with new phenomena. In this paper we extend a well-knownresult of Chang and Marshall on exponential integrability ofanalytic functions in the disk, to the case of quasiregularmappings defined in the unit ball of ⁿ. To this end, an ‘egg-yolk’principle is first established for such maps, which extendsa recent result of the first author. Our work leaves open aninteresting problem regarding n-harmonic functions.