The Heat Flow of the CCR Algebra

Let Pf(x) = –if'(x) and Qf(x) = xf(x) be the canonicaloperators acting on an appropriate common dense domain in L²(R).The derivations D_P(A) = i(PA–AP) and D_Q(A) = i(QA–AQ)act on the *-algebra A of all integral operators having smoothkernels of compact support, for example, and one may considerthe noncommutative ‘Laplacian’, L = + , as a linear mapping of A into itself. L generates a semigroup of normal completely positive linearmaps on B(L₂(R)), and this paper establishes some basic propertiesof this semigroup and its minimal dilation to an E₀-semigroup.In particular, the author shows that its minimal dilation ispure and has no normal invariant states, and he discusses thesignificance of those facts for the interaction theory introducedin a previous paper. There are similar results for the canonical commutation relationswith n degrees of freedom, where 1 n < . 2000 MathematicsSubject Classification 46L57 (primary), 46L53, 46L65 (secondary).     

A Characterization of Fredholm Pseudo-Differential Operators

We give a necessary and sufficient condition on an ellipticsymbol of order m to ensure that the unique closed extensionin L^p(Rⁿ) for 1 < p < , of the pseudo-differential operatorT, initially defined on the Schwartz space, is a Fredholm operatorfrom L^p(Rⁿ) into L^p(Rⁿ) with domain H^{m, p}, where H^{m, p} is theL^p Sobolev space of order m.     

The Second Dual Algebra of the Measure Algebra of a Compact Group

It is shown that for every compact group G, L¹(G)^ is uniqueand minimal among all the closed subsets I of M(G)** such thatI is a proper (0, M(G)**) algebraic ideal, and such that I issolid with respect to absolute continuity; that is, n L¹(G)^whenever n M(G)** and n << µ L¹(G)^. 1991 MathematicsSubject Classification 43A20, 43A22.     

Damping Oscillatory Integrals with Polynomial Phase and Convolution Operators with the Affine Arclength Measure on Polynomial Curves in Rn

McMichael proved that the convolution with the (euclidean) arclengthmeasure supported on the curve t (t, t², ..., tⁿ), 0 < t< 1, maps L^p(Rⁿ) boundedly into L^p'(Rⁿ) if and only if 2n(n+1)/(n²+n+2) p 2. In proving this, a uniform estimate on damping oscillatoryintegrals with polynomial phase was crucial. In this paper,a remarkably simple proof of the same estimate on oscillatoryintegrals is presented. In addition, it is shown that the convolutionoperator with the affine arclength measure on any polynomialcurve in Rⁿ maps L^p(Rⁿ) boundedly into L^p'(Rⁿ) if p = 2n(n+1)/(n²+n+2).     

Lie Powers of Modules for Groups of Prime Order

Let L(V) be the free Lie algebra on a finite-dimensional vectorspace V over a field K, with homogeneous components Lⁿ(V) forn 1. If G is a group and V is a KG-module, the action of Gextends naturally to L(V), and the Lⁿ(V) become finite-dimensionalKG-modules, called the Lie powers of V. In the decompositionproblem, the aim is to identify the isomorphism types of indecomposableKG-modules, with their multiplicities, in unrefinable directdecompositions of the Lie powers. This paper is concerned withthe case where G has prime order p, and K has characteristicp. As is well known, there are p indecomposables, denoted hereby J₁,...,J_p, where J_r has dimension r. A theory is developedwhich provides information about the overall module structureof LV) and gives a recursive method for finding the multiplicitiesof J₁,...,J_p in the Lie powers Lⁿ(V). For example, the theoryyields decompositions of L(V) as a direct sum of modules isomorphiceither to J₁ or to an infinite sum of the form J_r J_{p-1} J_{p-1} ... with r 2. Closed formulae are obtained for the multiplicitiesof J₁,..., J_p in Lⁿ(J_p and Lⁿ(J_{p-1). For r < p-1, the indecomposableswhich occur with non-zero multiplicity in Lⁿ(J_r) are identifiedfor all sufficiently large n. 2000 Mathematical Subject Classification:17B01, 20C20.     

Ko-Rings of S2k+1/Zz2n

For 2, we describe the reduced real K-theory K~O(L_k2ⁿ)) of the standard lens space L_k(2ⁿ)xs in terms of generators and relations. In particular, we show that the K~O-order of the Hopf bundle is 2^n–1+2[/2].     

Orthogonal Spaces Associated with Exponentials of Indicator Functions on Fock Space

Parthasarathy and Sunder have proved that the set of coherentvectors associated with the indicator functions of Borel setsis total in the boson Fock space (L²(R⁺;C)). The paper studiesthe space generated by coherent vectors associated with theunion of n intervals. A complete characterization is given oftheir orthogonal space in terms of their chaos expansion. Parthasarathyand Sunder's result is recovered in a very simple way. In thecases of the Brownian motion or Poisson process interpretationof the Fock space, the result characterizes those random variablesthat are orthogonal to the exponential of any sum of n incrementsof the Brownian motion or Poisson process.     

On the Volume Ratio of Two Convex Bodies

Let K and L be two convex bodies in Rⁿ. The volume ratio vr(K,L) of K and L is defined by vr(K, L = inf(|K|/|T(L)|)^1/n, wherethe infimum is over all affine transformations T of Rⁿ for whichT(L) K. It is shown in this paper that vr(K, L) , where c > 0 is an absolute constant. This isoptimal up to the logarithmic term. 2000 Mathematics SubjectClassification 52A40, 46B07 (primary); 52A21, 52A20 (secondary).     

Distributed control for cooperative systems involving parabolic operators with an infinite number of variables

^** Email: serraghm{at}yahoo.com The optimal control for cooperative systems involving parabolicoperators with an infinite number of variables is considered.First the existence and uniqueness of the states are proved;then the necessary and sufficient condition for the controlto be optimal is obtained by a set of inequalities. The controlin our problems is of distributed type and is allowed to bein the Hilbert space (L²(0, T, L²()))ⁿ.     

A Graph-Theoretic Approach to the Unique Midset Property of Metric Spaces

A metric space X has the unique midset property if there isa topology-preserving metric d on X such that for every pairof distinct points x, y there is one and only one point p suchthat d(x, p) = d(y, p). The following are proved. (1) The discretespace with cardinality n has the unique midset property if andonly if n 2, 4 and n c, where c is the cardinality of thecontinuum. (2) If D is a discrete space with cardinality notgreater than c, then the countable power D^N of D has the uniquemidset property. In particular, the Cantor set and the spaceof irrational numbers have the unique midset property. A finite discrete space with n points has the unique midsetproperty if and only if there is an edge colouring of the completegraph K_n such that for every pair of distinct vertices x, ythere is one and only one vertex p such that (xp) = (yp). Letump(K_n) be the smallest number of colours necessary for sucha colouring of K_n. The following are proved. (3) For each k 0, ump(K_2k+1) = k. (4) For each k 3, k ump(K_2k) 2k–1.     

Intertwining Maps from Certain Group Algebras

In [17, 18, 19], we began to investigate the continuity propertiesof homomorphisms from (non-abelian) group algebras. Alreadyin [19], we worked with general intertwining maps [3, 12]. Thesemaps not only provide a unified approach to both homomorphismsand derivations, but also have some significance in their ownright in connection with the cohomology comparison problem [4]. The present paper is a continuation of [17, 18, 19]; this timewe focus on groups which are connected or factorizable in thesense of [26]. In [26], G. A. Willis showed that if G is a connectedor factorizable, locally compact group, then every derivationfrom L¹(G) into a Banach L¹(G)-module is automatically continuous.For general intertwining maps from L¹(G), this conclusion isfalse: if G is connected and, for some nN, has an infinite numberof inequivalent, n-dimensional, irreducible unitary representations,then there is a discontinuous homomorphism from L¹(G into aBanach algebra by [18, Theorem 2.2] (provided that the continuumhypothesis is assumed). Hence, for an arbitrary intertwiningmap from L¹(G), the best we can reasonably hope for is a resultasserting the continuity of on a ‘large’, preferablydense subspace of L¹(G). Even if the target space of is a Banachmodule (which implies that the continuity ideal I() of is closed),it is not a priori evident that is automatically continuous:the proofs of the automatic continuity theorems in [26] relyon the fact that we can always confine ourselves to restrictionsto L¹(G) of derivations from M(G) [25, Lemmas 3.1 and 3.4].It is not clear if this strategy still works for an arbitraryintertwining map from L¹(G) into a Banach L¹(G)-module.     

On the L1 Mean of the Exponential Sum Formed with the Mobius Function

In this paper we study the L¹ mean (1) of the exponential sum M()=_nXµ(n)e(n), where µ(n)is the Möbius function and e(x)=e^2ix. From the Cauchy–Schwarzinequality and Parseval's identity, we have , (2) and it is an interesting problem to investigate whether (2)reflects the true order of magnitude of (1).     

The Pontryagin Rings of Moduli Spaces of Arbitrary Rank Holomorphic Bundles Over a Riemann Surface

The cohomology of M(n, d), the moduli space of stable holomorphicbundles of coprime rank n and degree d and fixed determinant,over a Riemann surface of genus g 2, has been widely studiedfrom a wide range of approaches. Narasimhan and Seshadri [17]originally showed that the topology of M(n, d) depends onlyon the genus g rather than the complex structure of . An inductivemethod to determine the Betti numbers of M(n, d) was first givenby Harder and Narasimhan [7] and subsequently by Atiyah andBott [1]. The integral cohomology of M(n, d) is known to haveno torsion [1] and a set of generators was found by Newstead[19] for n = 2, and by Atiyah and Bott [1] for arbitrary n.Much progress has been made recently in determining the relationsthat hold amongst these generators, particularly in the ranktwo, odd degree case which is now largely understood. A setof relations due to Mumford in the rational cohomology ringof M(2, 1) is now known to be complete [14]; recently severalauthors have found a minimal complete set of relations for the‘invariant’ subring of the rational cohomology ofM(2, 1) [2, 13, 20, 25]. Unless otherwise stated all cohomology in this paper will haverational coefficients.     

Best Constants for Uncentred Maximal Functions

We precisely evaluate the operator norm of the uncentred Hardy–Littlewoodmaximal function on L^p(R¹). Consequently, we compute the operatornorm of the ‘strong’ maximal function on L^p(Rⁿ),and we observe that the operator norm of the uncentred Hardy–Littlewoodmaximal function over balls on L^p(Rⁿ) grows exponentially asn. 1991 Mathematics Subject Classification 42B25.     

The Hardy Operator and the Gap Between L{infty} and BMO

We study boundedness and compactness properties of the Hardyintegral operator from a weighted Banach function space X(v) into L and BMO. We give a new simplecharacterization of compactness of T from X(v) into BMO. Weconstruct examples of spaces X(v) such that T(X(v)) is (a) boundedin L but not compact in BMO; (b) compact in BMO but not boundedin L; (c) bounded in BMO but neither bounded in L nor compactin BMO; (d) bounded in L, compact in BMO and yet not compactin L. In order to obtain the last of the counterexamples weconstruct a new weighted Banach function space.     

Sharp Results Concerning the Expression of Functions as Sums of Finite Differences

Let * denote convolution and let _x denote the Dirac measureat a point x. A function in L²(R)) is called a difference oforder 1 if it is of the form g-_x * g for some x R and g L²(R)).Also, a difference of order 2 is a function of the form for some x R and g L²(R)). In fact,the concept of a ‘difference of order s’ may bedefined in a similar manner for each s 0. If f denotes the Fouriertransform of f, it is known that a function f in L²(R)) is afinite sum of differences of order s if and only if , and the vector space of all suchfunctions is denoted by D_s (L²(R)). Every function in D_s (L²(R))is a sum of int(2s) + 1 differences of order s, where int(t)denotes the integer part of t. Thus, every function in D₁ (L²(R))is a sum of three first order differences, but it was provedin 1994 that there is a function in D₁ (L(R)) which is neverthe sum of two first order differences. This complemented, forthe group R, the corresponding result for first order differencesobtained by Meisters and Schmidt in 1972 for the circle group.The results show that there is a function in L₂ R such that,for each s 1/2, this function is a sum of int (2s) + 1 differencesof order s but it is never the sum of int (2s) differences oforder s. The proof depends upon extending to higher dimensionsthe following result in two dimensions obtained by Schmidt in1972 in connection with Heilbronn's problem: if x₁, x__n arepoints in the unit square, Following on from the work of Meisters and Schmidt, this workfurther develops a connection between certain estimates in combinatorialgeometry and some questions of sharpness in harmonic analysis.2000 Mathematics Subject Classification 42A38 (primary), 52A40(secondary).     

Norm attaining polynomials

If L 0 is a continuous symmetric n-linear form on a Banachspace and is the associatedcontinuous n-homogeneous polynomial, the ratio always lies between 1 and nⁿ/n!. At one extreme,if L is defined on Hilbert space, then . If L attains norm on Hilbert space, then also attains norm; in this case, we give anexplicit construction to provide a unit vector x₀ with . At the other extreme, if and L attains norm, then attains norm. We prove that in general the converseis not true.     

Anzai Skew Products with Lebesgue Component of Infinite Multiplicity

Let f be a 1-periodic C¹-function whose Fourier coefficientssatisfy the condition _n|n|³|f(n|² < . For every R\Q andm Z\{0}, we consider the Anzai skew product T(x, y) = (x +, y + mx + f(x)) acting on the 2-torus. It is shown that T hasinfinite Lebesgue spectrum on the orthocomplement L²(dx) ofthe space of functions depending only on the first variable.This extends some earlier results of Kushnirenko, Choe, Lemaczyk,Rudolph, and the author. 1991 Mathematics Subject Classification28D05.     

Lorentz Spaces Of Vector-Valued Measures

Given a non-atomic, finite and complete measure space (,,µ)and a Banach space X, the modulus of continuity for a vectormeasure F is defined as the function _F(t) = sup_µ(E)t |F|(E)and the space V^p,q(X) of vector measures such that t^–1/p'_F(t) L^q((0,µ()],dt/t) is introduced. It is shown thatV^p,q(X) contains isometrically L^p,q(X) and that L^p,q(X) = V^p,q(X)if and only if X has the Radon–Nikodym property. It isalso proved that V^p,q(X) coincides with the space of cone absolutelysumming operators from L^p',q^' into X and the duality V^p,q(X^*)=(Lp^',q^'(X))^*where 1/p+1/p^'= 1/q+1/q^' = 1. Finally, V^p,q(X) is identifiedwith the interpolation space obtained by the real method (V¹(X),V(X))_1/p^',q. Spaces where the variation of F is replaced bythe semivariation are also considered.