Higher string topology on general spaces

In this paper, I give a generalized analogue of the string topologyresults of Chas and Sullivan, and of Cohen and Jones. For afinite simplicial complex X and k 1, I construct a spectrumMaps(S^k, X)^S(X), which is obtained by taking a generalizationof the Spivak bundle on X (which however is not a stable spherebundle unless X is a Poincaré space), pulling back toMaps(S^k, X) and quotienting out the section at infinity. I showthat the corresponding chain complex is naturally homotopy equivalentto an algebra over the (k + 1)-dimensional unframed little diskoperad C_{k + 1}. I also prove a conjecture of Kontsevich, whichstates that the Quillen cohomology of a based C_k-algebra (inthe category of chain complexes) is equivalent to a shift ofits Hochschild cohomology, as well as prove that the operadC_*C_k is Koszul-dual to itself up to a shift in the derived category.This gives one a natural notion of (derived) Koszul dual C_*C_k-algebras.I show that the cochain complex of X and the chain complex of^k X are Koszul dual to each other as C_*C_k-algebras, and thatthe chain complex of Maps(S^k, X)^S(X) is naturally equivalentto their (equivalent) Hochschild cohomology in the categoryof C_* C_k-algebras. 2000 Mathematics Subject Classification 55P48(primary), 16E40, 55N45, 18D50 (secondary).     

Nest Representations of Directed Graph Algebras

This paper is a comprehensive study of the nest representationsfor the free semigroupoid algebra L_G of a countable directedgraph G as well as its norm-closed counterpart, the tensor algebraT⁺(G). We prove that the finite-dimensional nest representations separatethe points in L_G, and a fortiori, in T⁺(G). The irreduciblefinite-dimensional representations separate the points in L_Gif and only if G is transitive in components (which is equivalentto being semisimple). Also the upper triangular nest representationsseparate points if and only if for every vertex x T(G) supportinga cycle, x also supports at least one loop edge. We also study faithful nest representations. We prove that L_G(or T⁺(G) admits a faithful irreducible representation if andonly if G is strongly transitive as a directed graph. More generally,we obtain a condition on G which is equivalent to the existenceof a faithful nest representation. We also give a conditionthat determines the existence of a faithful nest representationfor a maximal type N nest. 2000 Mathematics Subject Classification47L80, 47L55, 47L40.     

The Natural Morphisms between Toeplitz Algebras on Discrete Groups

Let G be a discrete group and (G, G₊) be a quasi-ordered group.Set G₊(G₊)^–1 and G₁= (G₊\){e}. Let F^G₁(G) andF^G₊(G) be the corresponding Toeplitz algebras. In the paper,a necessary and sufficient condition for a representation ofF^G₊(G) to be faithful is given. It is proved that when G isabelian, there exists a natural C*-algebra morphism from F^G₁(G)to F^G₊(G). As an application, it is shown that when G = Z² andG₊ = Z₊ x Z, the K-groups K₀(F^G₁(G)) Z², K₁(F^G₁(G)) Z andall Fredholm operators in F^G₁(G) are of index zero.     

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.     

The Transformation Groups Whose C*-Algebras are Fell Algebras

Let (G, X) be a locally compact transformation group in whichG acts freely on X. We show that the associated transformation-groupC*-algebra C₀(X) G is a Fell algebra if and only if X is aCartan G-space.     

Isomorphisms of Group Algebras

Let G₁ and G₂ be locally compact groups. If T is an algebraisomorphism of L¹(G₁) onto L¹(G₂) with ||T|| (1+3), then G₁and G₂ are isomorphic. This improves on earlier results, and,in a certain sense, is best possible. However, the main conjecturethat the groups are isomorphic if ||T|| < 2 remains unsolvedexcept for abelian groups and for connected groups. Similarresults are given for the measure algebra M(G) and for the algebraC(G) of continuous functions when the group G is compact.     

Finite Time Equipartition for Second-order Hyperbolic Systems

A sufficient condition for equipartition of energy for secondorder hyperbolic systems in three space variables is given.The condition states that the system should evolve in such away that the time derivative of a solution of the form (u₁,0)^T is connected with the space derivatives of a solution ofthe form (0, u₂)^T and the time derivative of (0, u₂)^T is connectedwith the space derivatives of (u₁, 0)^T.     

From Endomorphisms to Automorphisms and Back: Dilations and Full Corners

When S is a discrete subsemigroup of a discrete group G suchthat G = S^–1S, it is possible to extend circle-valuedmultipliers from S to G, to dilate (projective) isometric representationsof S to (projective) unitary representations of G, and to dilate/extendactions of S by injective endomorphisms of a C*-algebra to actionsof G by automorphisms of a larger C*-algebra. These dilationsare unique provided they satisfy a minimality condition. The(twisted) semigroup crossed product corresponding to an actionof S is isomorphic to a full corner in the (twisted) crossedproduct by the dilated action of G. This shows that crossedproducts by semigroup actions are Morita equivalent to crossedproducts by group actions, making powerful tools available tostudy their ideal structure and representation theory. The dilationof the system giving the Bost–Connes Hecke C*-algebrafrom number theory is constructed explicitly as an application:it is the crossed product C₀(A_f)Q*₊, corresponding to the multiplicativeaction of the positive rationals on the additive group A_f offinite adeles.     

Smoothness Properties of the Unit Ball in a JB*-Triple

An element a of norm one in a JB^*-triple A is said to be smoothif there exists a unique element x in the unit ball A₁^* of thedual A^* of A at which a attains its norm, and is said to beFréchet-smooth if, in addition, any sequence (x_n) ofelements in A₁^* for which (x_n(a)) converges to one necessarilyconverges in norm to x. The sequence (a²ⁿ⁺¹) of odd powers ofa converges in the weak^*-topology to a tripotent u(a) in theJBW^*-envelope A^** of A. It is shown that a is smooth if andonly if u(a) is a minimal tripotent in A^** and a is Fréchet-smoothif and only if, in addition, u(a) lies in A.     

On the Local and Superlinear Convergence of Quasi-Newton Methods

This paper presents a local convergence analysis for severalwell-known quasi-Newton methods when used, without line searches,in an iteration of the form to solve for x^* such that Fx^* = 0. The basic idea behind theproofs is that under certain reasonable conditions on x_o, Fand x_o, the errors in the sequence of approximations {H_k} toF'(x^*)^–1 can be shown to be of bounded deterioration inthat these errors, while not ensured to decrease, can increaseonly in a controlled way. Despite the fact that H_k is not shownto approach F'(x^*)^–1, the methods considered, includingthose based on the single-rank Broyden and double-rank Davidon-Fletcher-Powellformulae, generate locally Q-superlinearly convergent sequences{x_k}.     

Quantum Markov Processes with a Christensen-Evans Generator in a Von Neumann Algebra

Let A be a unital von Neumann algebra of operators on a complexseparable Hilbert space H₀, and let {T_t, t 0} be a uniformlycontinuous quantum dynamical semigroup of completely positiveunital maps on A. The infinitesimal generator L of {T_t} is abounded linear operator on the Banach space A. For any Hilbertspace K, denote by B(K) the von Neumann algebra of all boundedoperators on K. Christensen and Evans [3] have shown that Lhas the form [formula] where is a representation of A in B(K) for some Hilbert spaceK, R: H₀ K is a bounded operator satisfying the ‘minimality’condition that the set {(RX–(X)R)u, uH₀, XA} is totalin K, and K₀ is a fixed element of A. The unitality of {T_t}implies that L(1) = 0, and consequently K₀=iH–R^*R, whereH is a hermitian element of A. Thus (1.1) can be expressed as [formula] We say that the quadruple (K, , R, H) constitutes the set ofChristensen–Evans (CE) parameters which determine theCE generator L of the semigroup {T_t}. It is quite possible thatanother set (K', ', R', H') of CE parameters may determine thesame generator L. In such a case, we say that these two setsof CE parameters are equivalent. In Section 2 we study thisequivalence relation in some detail. 1991 Mathematics SubjectClassification 81S25, 60J25.     

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 .     

Rank Distributions on Coarse Spaces and Ideal Structure of Roe Algebras

The notions of controlled truncations for operators in the Roealgebras C^* (X) of a coarse space (X, ) with uniformly locallyfinite coarse structure, and rank distributions on (X, ) areintroduced. It is shown that the controlled propagation operatorsin an ideal I of C^* (X) are exactly the controlled truncationsof elements in I. It follows that the lattice of the idealsof C^* (X) in which controlled propagation operators are denseis isomorphic to the lattice of all rank distributions on (X,). If X is a discrete metric space with Yu's property A, thenthe ideal structure of the Roe algebra C^* (X is completely determinedby the rank distributions on X. 2000 Mathematics Subject Classification46L80, 46L89.     

On Translation Functors for General Linear and Symmetric Groups

In this paper we study the rational representation theory ofthe general linear group G = GL_n(F) over an algebraically closedfield F of characteristic p. Given Z/pZ, we define functorsTr and Tr, which, roughly speaking, are given by tensoring withthe natural G-module V and its dual V* respectively, and thenprojecting onto certain blocks determined by the residue . Infact, these functors can be viewed as special cases of Jantzen'stranslation functors. We prove a number of fundamental propertiesabout these functors and also certain closely related functorsthat arise in the modular representation theory of the symmetricgroup. 1991 Mathematics Subject Classification: 20G05, 20C05.     

Linear Independence of Hecke Operators in the Homology of X0(N)

In Merel's recent proof [7] of the uniform boundedness conjecturefor the torsion of elliptic curves over number fields, a keystep is to show that for sufficiently large primes N, the Heckeoperators T₁, T₂, ..., T_D are linearly independent in theiractions on the cycle e from 0 to i in H₁(X₀(N) (C), Q). In particular,he shows independence when max(D⁸, 400D⁴) < N/(log N)⁴. Inthis paper we use analytic techniques to show that one can chooseD considerably larger than this, provided that N is large.     

The Singularities of Quantum Groups

If C[G] C[H] is an extension of Hopf domains of degree d, thenH G is an étale map. Equivalently, the variety X^{C}[H]of d-dimensional C[H]-modules compatible with the trace mapof the extension, is a smooth GL_d-variety with quotient G. Ifwe replace C[H] by a non-commutative Hopf algebra H, we constructsimilarly a GL^d-variety and quotient map : X_H G. The smoothlocus of H over C[G] is the set of points g G such that X_His smooth along {^-1}(g). We relate this set to the separabilitylocus of H over C[G] as well as to the (ordinary) smooth locusof the commutative extension C[G] Z where Z is the centre ofH. In particular, we prove that the smooth locus coincides withthe separability locus whenever H is a reflexive Azumaya algebra.This implies that the quantum function algebras O(G) and quantisedenveloping algebras U{g} are as singular as possible. 1991 MathematicsSubject Classification: 16W30, 16R30.     

POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

We study the relation between the polynomial numerical indicesof a complex vector-valued function space and the ones of itsrange space. It is proved that the spaces C(K, X) and L(µ,X) have the same polynomial numerical index as the complex Banachspace X for every compact Hausdorff space K and every -finitemeasure µ, which does not hold any more in the real case.We give an example of a complex Banach space X such that, forevery k 2, the polynomial numerical index of order k of X isthe greatest possible, namely 1, while the one of X** is theleast possible, namely k^k/(1–k). We also give new examplesof Banach spaces with the polynomial Daugavet property, namelyL(µ, X) when µ is atomless, and C_w(K, X), C_w*(K,X*) when K is perfect.     

On a Topological Property of certain Calkin Algebras

Let X = 1^p, 1 p < , or X = c₀, B(X) be the algebra of allbounded linear operators on X, H(X) be the ideal of compactoperators in B(X), and C(X) = B(X)/H(X) be the Calkin algebraon X. For TB(X), let ||T||_c = dist(T, H(X)) be the essentialnorm of T that is the norm of T+H(X) in C(X). It is shown thatfor any operator TB(X) and any number 0 < t < 1, thereexists a closed infinite dimensional subspace Z Z X such that ||Tx|| t||T||_c, for all x Z. As a consequence, it is shown that every (not necessarily complete)submultiplicative norm on the Calkin algebra C(X) is equivalentto the quotient norm || ||_c on C(X).     

Serre's Theorem on the Cohomology Algebra of a p-Group

The purpose of this note is to give a proof of a theorem ofSerre, which states that if G is a p-group which is not elementaryabelian, then there exist an integer m and non-zero elementsx₁, ..., x_m H¹ (G, Z/p) such that with ß the Bockstein homomorphism. Denote by m_G thesmallest integer m satisfying the above property. The theoremwas originally proved by Serre [5], without any bound on m_G.Later, in [2], Kroll showed that m_G p^k – 1, with k =dim_Z/pH¹ (G, Z/p). Serre, in [6], also showed that m_G (p^k –1)/(p – 1). In [3], using the Evens norm map, Okuyamaand Sasaki gave a proof with a slight improvement on Serre'sbound; it follows from their proof (see, for example, [1, Theorem4.7.3]) that m_G (p + 1)p^k–2. However, m_G can be sharpenedfurther, as we see below. For convenience, write H^*(G, Z/p) = H^*(G). For every x_i H¹(G),set 1991 Mathematics SubjectClassification 20J06.     

Mazur Intersection Properties and Differentiability of Convex Functions in Banach Spaces

It is proved that the dual of a Banach space with the Mazurintersection property is almost weak* Asplund. Analogously,the predual of a dual space with the weak* Mazur intersectionproperty is almost Asplund. Through the use of these arguments,it is found that, in particular, almost all (in the Baire sense)equivalent norms on l₁() and l() are Fréchet differentiableon a dense G subset. Necessary conditions for Mazur intersectionproperties in terms of convex sets satisfying a Krein–Milmantype condition are also discussed. It is also shown that, ifa Banach space has the Mazur intersection property, then everysubspace of countable codimension can be equivalently renormedto satisfy this property.