One-sided ideals and right cancellation in the second dual of the group algebra and similar algebras

The following results are proved for a non-compact, locallycompact group G: the dimension of every non-trivial right idealin L¹(G)** (equipped with the first Arens product) is at least, where (G) is the minimalnumber of compact sets required to cover G; there exist left ideals in L¹(G)** and in LUC(G)* with trivialintersections, and the linear span of right-cancellable elementsis weak*-dense in the annihilator of C₀(G) in LUC(G)* and inthe annihilator of (theL-functions that vanish at infinity) in L(G)*. The same resultsare proved for weighted algebras when the weight function isdiagonally bounded.     

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.     

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 .     

Nonpositively Curved Metric in the Positive Cone of a Finite Von Neumann Algebra

In this paper we study the metric geometry of the space ofpositive invertible elements of a von Neumann algebra A witha finite, normal and faithful tracial state . The trace inducesan incomplete Riemannian metric x,y_a = (ya^–1xa^–1),and, though the techniques involved are quite different, thesituation here resembles in many relevant aspects that of then x n matrices when they are regarded as a symmetric space.For instance, we prove that geodesics are the shortest pathsfor the metric induced, and that the geodesic distance is aconvex function; we give an intrinsic (algebraic) characterizationof the geodesically convex submanifolds M of ; and under a suitablehypothesis we prove a factorization theorem for elements inthe algebra that resembles the Iwasawa decomposition for matrices.This factorization is obtained via a nonlinear orthogonal projection_M : M, a map which turns out to be contractive for the geodesicdistance.     

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].     

Immanant Inequalities, Induced Characters, and Rank Two Partitions

If = {₁, ₂, ..., _s}, where _{1 2} ... _s > 0, is a partitionof n then denotes the associated irreducible character of S_n,the symmetric group on {1, 2, ..., n}, and, if cCS_n, the groupalgebra generated by C and S_n, then d_c(·) denotes thegeneralized matrix function associated with c. If c₁, c₂ CS_nthen we write c₁ c₂ in case (A) (A) for each n x n positivesemi-definite Hermitian matrix A. If cCS_n and c(e) 0, wheree denotes the identity in S_n, then or denotes (c(e))^–1 c. The main result, an estimate for the norms of tensors of a certainanti-symmetry type, implies that if = {₁, ₂, ..., _s, 1^t} isa partition of n such that s > 1 and _s = 2, and ' denotes{₁, ₂, ..., _s-1, 1^t} then (, _{2}) where denotes characterinduction from S_n–2 x S₂ to S_n. This in turn implies thatif = {₁, ₂, ..., _s, 1^t} with s > 1, _s = 2, and ßdenotes {₁ + 2, ₂, ..., _s-1, 1^t} then _ß which,in conjunction with other known results, provides many new inequalitiesamong immanants. In particular it implies that the permanentfunction dominates all normalized immanants whose associatedpartitions are of rank 2, a result which has proved elusivefor some years. We also consider the non-relationship problem for immanants– that is the problem of identifying pairs, (,ß)such that _ß and _ß are both false.     

Sub-Laplacians of Holomorphic Lp-Type on Exponential Solvable Groups

Let L denote a right-invariant sub-Laplacian on an exponential,hence solvable Lie group G, endowed with a left-invariant Haarmeasure. Depending on the structure of G, and possibly alsothat of L, L may admit differentiable L^p-functional calculi,or may be of holomorphic L^p-type for a given p 2. ‘HolomorphicL^p-type’ means that every L^p-spectral multiplier for Lis necessarily holomorphic in a complex neighbourhood of somenon-isolated point of the L²-spectrum of L. This can in factonly arise if the group algebra L¹(G) is non-symmetric. Assume that p 2. For a point in the dual g^* of the Lie algebrag of G, denote by ()=Ad^*(G) the corresponding coadjoint orbit.It is proved that every sub-Laplacian on G is of holomorphicL^p-type, provided that there exists a point g^* satisfying Boidol'scondition (which is equivalent to the non-symmetry of L¹(G)),such that the restriction of () to the nilradical of g is closed.This work improves on results in previous work by Christ andMüller and Ludwig and Müller in twofold ways: on theone hand, no restriction is imposed on the structure of theexponential group G, and on the other hand, for the case p>1,the conditions need to hold for a single coadjoint orbit only,and not for an open set of orbits. It seems likely that the condition that the restriction of ()to the nilradical of g is closed could be replaced by the weakercondition that the orbit () itself is closed. This would thenprove one implication of a conjecture by Ludwig and Müller,according to which there exists a sub-Laplacian of holomorphicL¹ (or, more generally, L^p) type on G if and only if there existsa point g^* whose orbit is closed and which satisfies Boidol'scondition.     

Movement and Separation of Subsets of Points Under Group Actions

Let G be a permutation group on a set , and let m and k be integerswhere 0<m<k. For a subset of , if the cardinalities ofthe sets ^g\, for gG, are finite and bounded, then is said tohave bounded movement, and the movement of is defined as move()=max_gG|^g\|. If there is a k-element subset such that move()m, it is shown that some G-orbit has length at most (k²–m)/(k–m).When combined with a result of P. M. Neumann, this result hasthe following consequence: if some infinite subset has boundedmovement at most m, then either is a G-invariant subset withat most m points added or removed, or nontrivially meets aG-orbit of length at most m²+m+1. Also, if move ()m for allk-element subsets and if G has no fixed points in , then either||k+m (and in this case all permutation groups on have thisproperty), or ||5m–2. These results generalise earlierresults about the separation of finite sets under group actionsby B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann,and groups in which all subsets have bounded movement (by theauthor).     

Real Interpolation and Two Variants of Gehring's Lemma

Let be a fixed open cube in Rⁿ. For r[1, ) and [0, ) we define where Q is a cube in Rⁿ (with sides parallel to the coordinateaxes) and _Q stands for the characteristic function of the cubeQ. A well-known result of Gehring [5] states that if (1.1) for some p(1, ) and c(0, ), then there exist q(p, ) and C=C(p,q, n, c)(0, ) such that for all cubes Q, where |Q| denotes the n-dimensional Lebesguemeasure of Q. In particular, a function fL¹() satisfying (1.1)belongs to L^q(). In [9] it was shown that Gehring's result is a particular caseof a more general principle from the real method of interpolation.Roughly speaking, this principle states that if a certain reversedinequality between K-functionals holds at one point of an interpolationscale, then it holds at other nearby points of this scale. Usingan extension of Holmstedt's reiteration formulae of [4] andresults of [8] on weighted inequalities for monotone functions,we prove here two variants of this principle involving extrapolationspaces of an ordered pair of (quasi-) Banach spaces. As an applicationwe prove the following Gehring-type lemmas.     

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 .     

The Full Muntz Theorem in C[0, 1] and L1[0, 1]

The main result of this paper is the establishment of the ‘fullMüntz Theorem’ in C[0, l]. This characterizes thesequences of distinct, positive real numbers for which span{l, x^₁, x^₂, ...} is dense in C[0,1]. The novelty of this result is the treatment of the mostdifficult case when inf_ii = 0 while sup_ii = . The paper settlesthe L and L₁ cases of the following. THEOREM (Full Müntz Theorem in L_p[0,1]). Let p [l, ].Suppose that is a sequence of distinct real numbers greater than –1/p. Then span{x^₀,x^₁, ...} is dense in L_p[0, 1] if and only if      

Normal Transversality and Uniform Bounds

Let A be a commutative ring. A graded A-algebra U = _n0 U_n isa standard A-algebra if U₀ = A and U = A[U₁] is generated asan A-algebra by the elements of U₁. A graded U-module F = _n0F_nis a standard U-module if F is generated as a U-module by theelements of F₀, that is, F_n = U_nF₀ for all n 0. In particular,F_n = U₁F_n–1 for all n 1. Given I, J, two ideals of A,we consider the following standard algebras: the Rees algebraof I, R(I) = _n0Iⁿtⁿ = A[It] A[t], and the multi-Rees algebraof I and J, R(I, J) = _n0(_p+q=nI^pJ^qu^pv^q) = A[Iu, Jv] A[u, v].Consider the associated graded ring of I, G(I) = R(I) A/I =_n0Iⁿ/Iⁿ⁺¹, and the multi-associated graded ring of I and J,G(I, J) = R(I, J) A/(I+J) = _n0(_p+q=nI^pJ^q/(I+J)I^pJ^q). We canalways consider the tensor product of two standard A-algebrasU = _p0U_p and V = _q0V_q as a standard A-algebra with the naturalgrading U V = _n0(_p+q=nU_p V_q). If M is an A-module, we havethe standard modules: the Rees module of I with respect to M,R(I; M) = _n0IⁿMtⁿ = M[It] M[t] (a standard R(I)-module), andthe multi-Rees module of I and J with respect to M, R(I, J;M) = _n0(_p+q=nI^pJ^qMu^pv^q) = M[Iu, Jv] M[u, v] (a standard R(I,J)-module). Consider the associated graded module of M withrespect to I, G(I; M) = R(I; M) A/I = _n0IⁿM/Iⁿ⁺¹M (a standardG(I)-module), and the multi-associated graded module of M withrespect to I and J, G(I, J; M) = R(I, J; M) A/(I+J) = _n0(_p+q=nI^pJ^qM/(I+J)I^pJ^qM)(a standard G(I, J)-module). If U, V are two standard A-algebras,F is a standard U-module and G is a standard V-module, thenF G = _n0(_p+q=nF_p G_q) is a standard U V-module. Denote by :R(I) R(J; M) R(I, J; M) and :R(I, J; M) R(I+J;M) the natural surjective graded morphisms of standard RI) R(J)-modules. Let :R(I) R(J; M) R(I+J; M) be . Denote by :G(I) G(J; M) G(I, J; M) and :G(I, J; M) G(I+J; M) the tensor productof and by A/(I+J); these are two natural surjective gradedmorphisms of standard G(I) G(J)-modules. Let :G(I) G(J; M) G(I+J; M) be . The first purpose of this paper is to prove the following theorem.     

Livsic Regularity Theorems for Twisted Cocycle Equations Over Hyperbolic Systems

Let be a hyperbolic map. Cocycle equations of the form f =u·g·u^–1 are considered, with f, g, u takingvalues in a compact connected Lie group G, being an automorphismof G and f, g being Hölder continuous. When the eigenvaluesof the derivative of have modulus 1, it is proved that anymeasurable solution u has a Hölder continuous version.This condition on is optimal. When f, g are C^k then u may betaken to be C^k–1+ for any (0, 1).     

Geometry of Critical Loci

Let :(Z,z)(U,0) be the germ of a finite (that is, proper with finite fibres)complex analytic morphism from a complex analytic normal surfaceonto an open neighbourhood U of the origin 0 in the complexplane C². Let u and v be coordinates of C² defined on U. Weshall call the triple (, u, v) the initial data. Let stand for the discriminant locus of the germ , that is,the image by of the critical locus of . Let ()_A be the branches of the discriminant locus at O whichare not the coordinate axes. For each A, we define a rational number d by where I(–, –) denotes the intersection number at0 of complex analytic curves in C². The set of rational numbersd, for A, is a finite subset D of the set of rational numbersQ. We shall call D the set of discriminantal ratios of the initialdata (, u, v). The interesting situation is when one of thetwo coordinates (u, v) is tangent to some branch of , otherwiseD = {1}. The definition of D depends not only on the choiceof the two coordinates, but also on their ordering. In this paper we prove that the set D is a topological invariantof the initial data (, u, v) (in a sense that we shall definebelow) and we give several ways to compute it. These resultsare first steps in the understanding of the geometry of thediscriminant locus. We shall also see the relation with thegeometry of the critical locus.     

Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rankn defined over the field k, where n3. Let be the root systemassociated with G(k) and let ={₁, ₂, ..., _n} be a set of fundamentalroots of , with ⁺ being the set of positive roots of with respectto . For and ⁺, let n() be the coefficient of in the expressionof as a sum of fundamental roots; so =n(). Also we recall thatht(), the height of , is given by ht()=n(). The highest rootin ⁺ will be denoted by . We additionally assume that the Dynkindiagram of G(k) is connected.     

BOUNDARY CONCENTRATION IN RADIAL SOLUTIONS TO A SYSTEM OF SEMILINEAR ELLIPTIC EQUATIONS

We study concentration phenomena for the system in the unit ball B₁ of ³ with Dirichlet boundaryconditions. Here , , > 0 and p > 1. We prove the existenceof positive radial solutions (, ) such that concentrates ata distance (/2)|log | away from the boundary B₁ as the parameter tends to 0. The approach is based on a combination of Lyapunov–Schmidtreduction procedure together with a variational method.     

Generators and defining relations for ring of invariants of commuting locally nilpotent derivations or automorphisms

Let A be an algebra over a field K of characteristic zero andlet ₁, ..., _sDer _K(A) be commuting locally nilpotent K-derivationssuch that _i(x_j) equals _ij, the Kronecker delta, for some elementsx₁, ..., x_sA. A set of generators for the algebra is found explicitly and a set of defining relationsfor the algebra A is described. Similarly, let ₁, ..., _s Aut_K(A)be commuting K-automorphisms of the algebra A is given suchthat the maps _i – id_A are locally nilpotent and _i (x_j)= x_j + _ij, for some elements x₁, ..., x_s A. A set of generatorsfor the algebra A: = {a A | ₁(a) = ... = _s(a) = a} is foundexplicitly and a set of defining relations for the algebra Ais described. In general, even for a finitely generated non-commutativealgebra A the algebras of invariants A and A are not finitelygenerated, not (left or right) Noetherian and a minimal numberof defining relations is infinite. However, for a finitely generatedcommutative algebra A the opposite is always true. The derivations(or automorphisms) just described appear often in many differentsituations (possibly) after localization of the algebra A.     

Random walks on free products of cyclic groups

Let G be a free product of a finite family of finite groups,with the set of generators being formed by the union of thefinite groups. We consider a transient nearest-neighbour randomwalk on G. We give a new proof of the fact that the harmonicmeasure is a special Markovian measure entirely determined bya finite set of polynomial equations. We show that in severalsimple cases of interest, the polynomial equations can be explicitlysolved to get closed form formulae for the drift. The examplesconsidered are /2 /3, /3 /3, /k /k and the Hecke groups /2 /k.We also use these various examples to study Vershik's notionof extremal generators, which is based on the relation betweenthe drift, the entropy and the growth of the group.     

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).