Some Remarks on the Cone of Completely Positive Maps between Von Neumann Algebras

The close relationship between the notions of positive formsand representations for a C*-algebra A is one of the most basicfacts in the subject. In particular the weak containment ofrepresentations is well understood in terms of positive forms:given a representation of A in a Hilbert space H and a positiveform on A, its associated representation is weakly containedin (that is, ker ker ) if and only if belongs to the weak*closure of the cone of all finite sums of coefficients of .Among the results on the subject, let us recall the followingones. Suppose that A is concretely represented in H. Then everypositive form on A is the weak* limit of forms of the typex ^k_{i=1 i}, x_i with the _i in H; moreover if A is a von Neumannsubalgebra of (H) and is normal, there exists a sequence (_i)_{i 1} in H such that (x) = _{i 1 i}, x_i for all x.     

A Torsion-Free Milnor-Moore Theorem

Let X be the space of Moore loops on a finite, q-connected,n-dimensional CW complex X, and let R Q be a subring containing1/2. Let (R) be the least non-invertible prime in R. For a gradedR-module M of finite type, let FM = M/Torsion M. We show thatthe inclusion P FH_*(X;R) of the sub-Lie algebra of primitiveelements induces an isomorphism of Hopf algebras provided that (R) n/q. Furthermore, the Hurewiczhomomorphism induces an embedding of F(_*(X) R) in P, with P/F(_*(X)R)torsion. As a corollary, if X is elliptic, then FH_*(X;R) isa finitely generated R-algebra.     

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).     

Constructing {kappa}-like Models of Arithmetic

A model (M, <, ...) is -like if M has cardinality but, forall M, the cardinality of {x M : x < a} is strictly lessthan . In this paper we shall give constructions of -like modelsof arithmetic satisfying an arbitrarily large finite part ofPA but not PA itself, for various singular cardinals . The mainresults are: (1) for each countable nonstandard M ₂–Th(PA)with arbitrarily large initial segments satisfying PA and eachuncountable of cofinality there is a cofinal extension K ofM which is -like; also hierarchical variants of this resultfor _n–Th(PA); and (2) for every n 1, every singular and every M B_n+exp+¬ I_n there is a -like model K elementarilyequivalent to M.     

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.     

Normal and Non-Normal Points of Self-Similar Sets and Divergence Points of Self-Similar Measures

Let K and µ be the self-similar set and the self-similarmeasure associated with an IFS (iterated function system) withprobabilities (S_i, p_i)_i=1,...,N satisfying the open set condition.Let ={1,...,N}^N denote the full shift space and let : K denotethe natural projection. The (symbolic) local dimension of µat is defined by lim_n (log µK_|n/log diam K_|n), where for = (₁, ₂,...) . A point for which the limit lim_n (log µK_|n/log diam K_|n) doesnot exist is called a divergence point. In almost all of theliterature the limit lim_n (log µK_|n/log diam K_|n) is assumedto exist, and almost nothing is known about the set of divergencepoints. In the paper a detailed analysis is performed of theset of divergence points and it is shown that it has a surprisinglyrich structure. For a sequence (_n)_n, let A(_n) denote the setof accumulation points of (_n)_n. For an arbitrary subset I ofR, the Hausdorff and packing dimension of the set and related sets is computed. An interesting and surprisingcorollary to this result is that the set of divergence pointsis extremely ‘visible’; it can be partitioned intoan uncountable family of pairwise disjoint sets each with fulldimension. In order to prove the above statements the theory of normaland non-normal points of a self-similar set is formulated anddeveloped in detail. This theory extends the notion of normaland non-normal numbers to the setting of self-similar sets andhas numerous applications to the study of the local propertiesof self-similar measures including a detailed study of the setof divergence points.     

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.     

Periodic Solutions of Lienard Equations with Asymmetric Nonlinearities at Resonance

The existence of 2-periodic solutions of the second-order differentialequation where a, b satisfy and p(t)=p(t+2),t R, is examined. Assume that limits lim_x±F(x)=F(±)(F(x)=) and lim_x±g(x)=g(±)exist and are finite. It is proved that the equation has atleast one 2-periodic solution provided that the zeros of thefunction ₁ are simple and the zeros of the functions ₁, ₂ aredifferent and the signs of ₂ at the zeros of ₁ in [0,2/n) donot change or change more than two times, where ₁ and ₂ aredefined as follows: Moreover, it is also proved that the given equation has at leastone 2-periodic solution provided that the following conditionshold: with 1 p < q 2.     

Sparse Systems of Functions Closed on Large Sets in RN

Let E(Z) = {e^inx}_nZ denote the trigonometrical exponential system.It is well known that E(Z) forms an orthogonal basis in thespace L²(0, 2). In 1964, H. Landau discovered that the trigonometricalsystem has the following property: certain small perturbationsof E(Z) yield exponential systems which are complete in L² onany finite union of 2-periodic translations of any interval(, 2–), 0 < < .     

Projective Modules over the Non-Commutative Sphere

The positive cone of the K₀-group of the non-commutative sphereB is explicitly determined by means of the four basic unboundedtrace functionals discovered by Bratteli, Elliott, Evans andKishimoto. The C*-algebra B is the crossed product A x Z₂ ofthe irrational rotation algebra A by the flip automorphism defined on the canonical unitary generators U, V by (U) = U*,(V) = V*, where VU = e²ⁱ UV and is an irrational real number.This result combined with Rieffel's cancellation techniquesis used to show that cancellation holds for all finitely generatedprojective modules over B. Subsequently, these modules are determinedup to isomorphism as finite direct sums of basic modules. Italso follows that two projections p and q in a matrix algebraover B are unitarily equivalent if, and only if, their vectortraces are equal: [p] = [q]. These results will have the following ramifications. They areused (elsewhere) to show that the flip automorphism on A isan inductive limit automorphism with respect to the basic buildingblock construction of Elliott and Evans for the irrational rotationalgebra. This will, in turn, yield a two-tower proof of thefact that B is approximately finite dimensional, first provedby Bratteli and Kishimoto.     

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.     

Calderon-Zygmund Operators on Mixed Lebesgue Spaces and Applications to Null Forms

The boundedness of Calderón–Zygmund operators isproved in the scale of the mixed Lebesgue spaces. As a consequence,the boundedness of the bilinear null forms Q_{i j} (u,) =_i u_j - _j u_i , Q₀(u,)=u_{t t} -_x u· _x on various space–timemixed Sobolev–Lebesgue spaces is shown.     

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.     

Picard Groups of Irrational Rotation C*-Algebras

Let be an irrational number in [0, 1] and A the correspondingirrational rotation C*-algebra. Let Aut (A) be the group ofall automorphisms of A and Int (A) the normal subgroup of Aut(A) of all inner automorphisms of A. Let Pic (A) be the Picardgroup of A. In the present note we shall show that if is notquadratic, then Pic (A)Aut (A)/Int (A) and that if is quadratic,then Pic (A) is isomorphic to a semidirect product of Aut (A)/Int(A) with Z. Furthermore, in the last section we shall discussPicard groups of certain Cuntz algebras.     

Faithful Representations of Free Products

In 1940 Nisnevi published the following theorem [3]. Let (G) be a family of groups indexed by some set and (F) a family of fields of the same characteristic p0. Iffor each the group G has a faithful representation of degreen over F then the free product* G has a faithful representationof degree n+1 over some field of characteristic p. In [6] Wehrfritzextended this idea. If (G) GL(n, F) is a family of subgroupsfor which there exists ZGL(n, F) such that for all the intersectionGF.1_n=Z, then the free product of the groups *_ZG with Z amalgamatedvia the identity map is isomorphic to a linear group of degreen over some purely transcendental extension of F. Initially, the purpose of this paper was to generalize theseresults from the linear to the skew-linear case, that is, togroups isomorphic to subgroups of GL(n, D) where the D are divisionrings. In fact, many of the results can be generalized to ringswhich, although not necessarily commutative, contain no zero-divisors.We have the following.     

Von Neumann Eta-Invariants and C*-Algebra K-Theory

Let M be a smooth, compact, oriented, odd-dimensional Riemannianmanifold and let M be anormal covering of M. It is proved that the relative von Neumanneta-invariant ₍₂₎() of Cheeger and Gromov is a homotopy invariant when is torsion-free, discreteand the Baum–Connes assembly map µ_max:K₀(B) K₀(C*)is an isomorphism.     

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||=+.     

The A-Polynomial has Ones in the Corners

1. Definition of the A-polynomial The A-polynomial was introduced in [3] (see also [5]), and wepresent an alternative definition here. Let M be a compact 3-manifoldwith boundary a torus T. Pick a basis , µ of ₁T, whichwe shall refer to as the longitude and meridian. Consider thesubset R_U of the affine algebraic variety R = Hom (₁M, SL₂C)having the property that () and (µ) are upper triangular.This is an algebraic subset of R, since one just adds equationsstating that the bottom-left entries in certain matrices arezero. There is a well-defined eigenvalue map given by taking the top-left entries of () and (µ).1991 Mathematics Subject Classification 57M25, 57M50.     

Identity Theorems for Functions of Bounded Characteristic

Suppose that f(z) is a meromorphic function of bounded characteristicin the unit disk :|z|<1. Then we shall say that f(z)N. Itfollows (for example from [3, Lemma 6.7, p. 174 and the following])that where h₁(z), h₂(z) are holomorphic in and have positive realpart there, while ₁(z), ₂(z) are Blaschke products, that is, where p is a positive integer or zero, 0<|a_j|<1, c isa constant and (1–|a_j|)<. We note in particular that, if c0, so that f(z)0, (1.1) so that f(z)=0 only at the points a_j. Suppose now that z_j isa sequence of distinct points in such that |z_j|1 as j and (1–|z_j|)=. (1.2) If f(z_j)=0 for each j and fN, then f(z)0. N. Danikas [1] has shown that the same conclusion obtains iff(z_j)0 sufficiently rapidly as j. Let _j, _j be sequences of positivenumbers such that _j< and _j as j. Danikas then defines and proves Theorem A.     

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).