Strength of convergence and multiplicities in the spectrum of a C*-dynamical system

We consider separable C*-dynamical systems (A, G,) for whichthe induced action of the group G on the primitive ideal spacePrim A of the C*-algebra A is free. We study how the representationtheory of the associated crossed product C*-algebra A G dependson the representation theory of A and the properties of theaction of G on Prim A and the spectrum Â. Our main toolsinvolve computations of upper and lower bounds on multiplicitynumbers associated to irreducible representations of A G. Weapply our techniques to give necessary and sufficient conditions,in terms of A and the action of G, for AG to be (i) a continuous-traceC*-algebra, (ii) a Fell C*-algebra and (iii) a bounded-traceC*-algebra. When G is amenable, we also give necessary and sufficientconditions for the crossed product C*-algebra AG to be (iv)a liminal C*-algebra and (v) a Type I C*-algebra. The resultsin (i), (iii)–(v) extend some earlier special cases inwhich A was assumed to have the corresponding property.     

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).     

A bifurcation problem governed by the boundary condition II

In this work we consider the problem u = a(x)u^p in on , where is a smooth bounded domain, isthe outward unit normal to , is regarded as a parameter and0 < p < 1. We consider both cases where a(x) > 0 in or a(x) is allowed to vanish in a whole subdomain ₀ of . Ourmain results include existence of non-negative non-trivial solutionsin the range 0 < < ₁, where ₁ is characterized by meansof an eigenvalue problem, uniqueness and bifurcation from infinityof such solutions for small , and the appearance of dead coresfor large enough .     

Scattering matrices and Weyl functions

For a scattering system {A, A₀} consisting of self-adjoint extensionsA and A₀ of a symmetric operator A with finite deficiency indices,the scattering matrix {S()} and a spectral shift function arecalculated in terms of the Weyl function associated with a boundarytriplet for A*, and a simple proof of the Krein–Birmanformula is given. The results are applied to singular Sturm–Liouvilleoperators with scalar and matrix potentials, to Dirac operatorsand to Schrödinger operators with point interactions.     

Grothendieck's Inequalities for Real and Complex JBW*-Triples

We prove that, if and >0, if V and W are complex JBW*-triples (with preduals V_* andW_*, respectively), and if U is a separately weak*-continuousbilinear form on V x W, then there exist norm-one functionals₁, ₂ V_* and ₁, ₂ W_* satisfying for all (x, y) V x W. Here, for a norm-one functional on acomplex JB*-triple V, |·| stands for the prehilbertianseminorm on V associated to given by for all x W, where z V^** satisfies z = |z| =1. We arrive at this form of ‘Grothendieck's inequality’through results of C.-H. Chu, B. Iochum, and G. Loupias, andan amended version of the ‘little Grothendieck's inequality’for complex JB*-triples due to T. Barton and Y. Friedman. Wealso obtain extensions of these results to the setting of realJB*-triples. 2000 Mathematical Subject Classification: 17C65,46K70, 46L05, 46L10, 46L70.     

Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball

Weighted Triebel–Lizorkin and Besov spaces on the unitball B^d in ^d with weights w_µ(x)=(1–|x|²)^µ–1/2,µ0, are introduced and explored. A decomposition schemeis developed in terms of almost exponentially localized polynomialelements (needlets) {}, {} and it is shown that the membershipof a distribution to the weighted Triebel–Lizorkin orBesov spaces can be determined by the size of the needlet coefficients{f, } in appropriate sequence spaces.     

Moduli of McKay quiver representations I: the coherent component

For a finite abelian group G GL (n, ), we describe the coherent component Y of the moduli space of-stable McKay quiver representations. This is a not-necessarily-normaltoric variety that admits a projective birational morphism obtained by variation of GeometricInvariant Theory quotient. As a special case, this gives a newconstruction of Nakamura's G-Hilbert scheme Hilb^G that avoidsthe (typically highly singular) Hilbert scheme of |G|-pointsin . To conclude, we describe the toric fan of Y and hence calculate the quiver representationcorresponding to any point of Y.     

Hopf C*-Algebras

In this paper we define and study Hopf C^*-algebras. Roughlyspeaking, a Hopf C^*-algebra is a C^*-algebra A with a comultiplication: A M(A A) such that the maps a b (a)(1 b) and a (a 1)(b)have their range in A A and are injective after being extendedto a larger natural domain, the Haagerup tensor product A _hA. In a purely algebraic setting, these conditions on are closelyrelated to the existence of a counit and antipode. In this topologicalcontext, things turn out to be much more subtle, but neverthelessone can show the existence of a suitable counit and antipodeunder these conditions. The basic example is the C^*-algebra C₀(G) of continuous complexfunctions tending to zero at infinity on a locally compact groupwhere the comultiplication is obtained by dualizing the groupmultiplication. But also the reduced group C^*-algebra of a locally compact group with thewell-known comultiplication falls in this category. In factall locally compact quantum groups in the sense of Kustermansand the first author (such as the compact and discrete ones)as well as most of the known examples are included. This theory differs from other similar approaches in that thereis no Haar measure assumed. 2000 Mathematics Subject Classification: 46L65, 46L07, 46L89.     

On the Fredholm Alternative for the p-Laplacian in One Dimension

We investigate the existence of a weak solution u to the quasilineartwo-point boundary value problem We assume that 1 < p < p ¬ = 2, 0 < a < , andthat f L¹(0,a) is a given function. The number _k stands forthe k-th eigenvalue of the one-dimensional p-Laplacian. Let_{p p} x/a) denote the eigenfunction associated with ₁; then _p(k_p x/a) is the eigenfunction associated with _k. We show the existenceof solutions to (P) in the following cases. (i) When k=1 and f satisfies the orthogonality condition the set of solutions is bounded. (ii) If k=1 and f_t L¹(0,a) is a continuous family parametrizedby t [0,1], with then there exists some t^* [0,1] such that (P) has a solutionfor f = f_t^*. Moreover, an appropriate choice of t_* yields asolution u with an arbitrarily large L¹(0,a)-norm which meansthat such f cannot be orthogonal to _pp x/a. (iii) When k 2 and f satisfies a set of orthogonality conditionsto _p(k p x/a) on the subintervals , again, the set of solutions is bounded. is a continuous family satisfying either or another related condition, then there exists some t^* [0,1]such that (P) has a solution for f = f_t*. Prüfer's transformation plays the key role in our proofs.2000 Mathematical Subject Classification: primary 34B16, 47J10;secondary 34L40, 47H30.     

Ideal Spaces of Banach Algebras

The ideal space Id(A) of a Banach algebra A is studied as abitopological space Id(A), _u, _n, where _u is the weakest topologyfor which all the norm functions I || a + I|| (with a A andI Id(A)) are upper semi-continuous, and _n is the de Groot dualof _u. When A is separable, _nu is either a compact, metrizabletopology, or it is neither Hausdorff nor first countable. TAF-algebrasare shown to exhibit the first type of behaviour. Applicationsto Banach bundles (which motivate the study), and to PI-Banachalgebras, are given. 1991 Mathematics Subject Classification:46H10, 46J20.     

The Tracial Topological Rank of C*-Algebras

We introduce the notion of tracial topological rank for C*-algebras.In the commutative case, this notion coincides with the coveringdimension. Inductive limits of C*-algebrasof the form PM_n(C(X))P,where X is a compact metric space with dim X k, and P is aprojection in M_n(C(X)), have tracial topological rank no morethan k. Non-nuclear C*-algebras can have small tracial topologicalrank. It is shown that if A is a simple unital C*-algebra withtracial topological rank k (< ), then

(i) A is quasidiagonal,

(ii) A has stable rank 1,

(iii) A has weakly unperforatedK₀(A),

(iv) A has the following Fundamental Comparabilityof Blackadar:if p, q A are two projections with (p) < (q)for all tracialstates on A, then p q

. 2000 MathematicsSubject Classification: 46L05, 46L35.     

Spectra of graph neighborhoods and scattering

Let (G)_>0 be a family of ‘-thin’ Riemannian manifoldsmodeled on a finite metric graph G, for example, the -neighborhoodof an embedding of G in some Euclidean space with straight edges.We study the asymptotic behavior of the spectrum of the Laplace–Beltramioperator on G, as 0, for various boundary conditions. We obtaincomplete asymptotic expansions for the kth eigenvalue and theeigenfunctions, uniformly for kC^–1, in terms of scatteringdata on a non-compact limit space. We then use this to determinethe quantum graph which is to be regarded as the limit object,in a spectral sense, of the family (G). Our method is a directconstruction of approximate eigenfunctions from the scatteringand graph data, and the use of a priori estimates to show thatall eigenfunctions are obtained in this way.     

The Ramifications of the Centres: Quantised Function Algebras at Roots of Unity

This paper continues the study of quantised function algebrasO[G] of a semisimple group G at an lth root of unity . Thesealgebras were introduced by De Concini and Lyubashenko in 1994,and studied further by De Concini and Procesi and by Gordon,amongst others. Our main purpose here is to increase understandingof the finite-dimensional factor algebras O[G](g), for g G.We determine the representation type and block structure ofthese factors, and (for many g) describe them up to isomorphism.A series of parallel results is obtained for the quantised Borelalgebras and . 2000 Mathematical Subject Classification: 16W35,17B37.     

The Approximation Numbers of Hardy-Type Operators on Trees

The Hardy operator T_a on a tree is defined by Properties of T_a as a map from L^p() into itselfare established for 1 p . The main result is that, with appropriateassumptions on u and v, the approximation numbers a_n(T_a) ofT_a satisfy for a specified constant _p and 1 p < . This extends results of Naimark, Newmanand Solomyak for p = 2. Hitherto, for p 2, (*) was unknowneven when is an interval. Also, upper and lower estimates forthe l^q and weak-l^q norms of a_n(T_a) are determined. 2000 MathematicalSubject Classification: 47G10, 47B10.     

Euler characteristics of the real points of certain varieties of algebraic tori

Let G be a complex connected reductive group which is definedover , let be its Lie algebra, and let be the variety of maximaltori of G. For (), let be the variety of tori in whose Liealgebra is orthogonal to with respect to the Killing form.We show, using the Fourier–Sato transform of conical sheaveson real vector bundles, that the ‘weighted Euler characteristic’of () is zero unless is nilpotent, in which case it equals(–1)^{(dim )/2}. Here ‘weighted Euler characteristic’means the sum of the Euler characteristics of the connectedcomponents, each weighted by a sign ± 1 which dependson the real structure of the tori in the relevant component.This is a real analogue of a result over finite fields whichis connected with the Steinberg representation of a reductivegroup.     

Almost Everywhere Convergence of Bochner-Riesz Means On The Heisenberg Group and Fractional Integration On The Dual

Let L denote the sub-Laplacian on the Heisenberg group H_n and the corresponding Bochner-Riesz operator. Let Q denote the homogeneous dimension and D the Euclideandimension of H_n. We prove convergence a.e. of the Bochner-Rieszmeans as r 0 for > 0and for all f L^p(H_n), provided that . Our proof is based on explicit formulas for the operators with a C, defined on the dual ofH_n by , which may be of independent interest. Here is given by for all (z,u) H_n. 2000 Mathematical Subject Classification: 22E30, 43A80.     

On Decomposition Numbers and Branching Coefficients for Symmetric and Special Linear Groups

In this paper we find the multiplicities dim L()_– where is an arbitrary root and L() is an irreducible SL_n-module withhighest weight . We provide different bases of the correspondingweight spaces and outline some applications to the symmetricgroups. In particular we describe certain composition multiplicitiesin the modular branching rule. 1991 Mathematics Subject Classification:20C05, 20G05.     

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

Let a_n be an increasing sequence of positive reals with a_n as n . Necessary and sufficient conditions are obtained foreach of the sequences to take on infinitely many prime values for almost all > rß.For example, the sequence a_n is infinitely often prime for almostall > 0 if and only if there is a subsequence of the a_n,say b_n, with b_{n + 1} > b_n + 1 and with the series divergent. Asymptotic formulae areobtained when the sequences considered are lacunary. An earlierresult of the author reduces the problem to estimating the measureof overlaps of certain sets, and sieve methods are used to obtainthe correct order upper bounds. 1991 Mathematics Subject Classification:primary 11N05; secondary 11K99, 11N36.