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.     

Some Compactness Results in Real Interpolation for Families of Banach Spaces

The paper shows that, if the operator T:A()B() is compact foralmost every , then is compact when or is the interpolation functor constructed for infinitefamilies of Banach spaces and S satisfies certain conditions.     

On the Norm of Elementary Operators

The norm problem is considered for elementary operators of theform U_a,b: AA,x axb+bxa (a,bA) in the special case when A isa subalgebra of the algebra of bounded operators on a Banachspace. In particular, the lower estimate || is established when the Banach space is a Hilbertspace and A is the algebra of all bounded linear operators.     

Barrelled Countable Enlargements and the Bounding Cardinal

Replacing c with the bounding cardinal improves two standardBCE results of Robertson, Tweddle and Yeomans, and is optimalfor the BCE codimension/inheritance result of Bonet and PérezCarreras. Indeed, is the smallest infinite-dimensionality formetrizable barrelled spaces, and is the largest cardinal suchthat every subspace of codimension less than in a metrizablebarrelled space is itself barrelled. We thus reconfirm as oneof two ‘optimal cardinals for metrizable barrelled spaces’.Esthetically pleasing, these properties of immediately solvethe normable BCE problem without extra-ZFC axiomatic assumptions,reduce the separable quotient problem to Banach spaces E with density character of E c, and are part of the solution toone version of the metrizable BCE problem found in our sequel,devoted to the other ‘optimal cardinal’.     

Multipliers on Banach spaces of functions on a locally compact Abelian group

Let E be a Banach space of functions on a locally compact Abeliangroup G satisfying certain conditions. It has been proved thatfor every bounded operator M on E commuting with translationsthere exists such that , where is a suitable subset of the group of the continuous morphismsfrom G into * and is a generalized Fourier transform of g defined on .     

Generating Countable Sets of Permutations

Let E be an infinite set. In answer to a question of Wagon,I show that every countable subset of the symmetric group Sym(E)is contained in a 2-generator subgroup of Sym(E). In answerto a question of Macpherson and Neumann, I show that, if Sym(E)is generated by A B where |B| ||E||, then Sym(E) is generatedby A {} for some permutation in Sym(E).     

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.     

Character Quotients for Coprime Acting Groups

Let the finite group A be acting on a finite group G with (|A|,|G|)=1. Let be the semidirect product of A and G. Let be acharacter of irreducible after restriction to G. In a previouspaper by Brian Hartley and the author, we proved that the restrictionof to S belongs to the set C(S) obtained by running over all that arise in this manner, by assuming, in addition, that Gis a product of extraspecial groups. This was proved in general,assuming only some condition on the Green functions of groupsof Lie type that is not as yet fully verified. In the presentpaper, we define the map Q(): SC by Q()(s)=|C_G(s)|/(s). We provethat Q()C(S) under the same hypotheses. In particular, the characterquotient Q() is an ordinary character.     

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.     

The Weiss Conjecture for Bounded Analytic Semigroups

New results concerning the so-called Weiss conjecture on admissibleoperators for bounded analytic semigroups are given. Let be a bounded analytic semigroup withgenerator –A on some Banach space X. It is proved thatif A^1/2 is admissible for A, that is, if there is an estimate then any continuous mappingC : D(A) Y valued in a Banach space Y is admissible for A providedthat there is an estimate .for , Re()<0. This holds in particular if is a contractive (analytic) semigroup on Hilbertspace. In the converse direction, it is shown that this mayhappen for a bounded analytic semigroup on Hilbert space thatis not similar to a contractive one. Applications in non-HilbertianBanach spaces are also given.     

Properties of Removable Singularities for Hardy Spaces of Analytic Functions

Removable singularities for Hardy spaces H^p() = {f Hol(): |f|^p u in for some harmonic u}, 0 < p < are studied. A setE = is a weakly removable singularity for H^p(\E) if H^p(\E) Hol(), and a strongly removable singularity for H^p(\E) if H^p(\E)= H^p(). The two types of singularities coincide for compactE, and weak removability is independent of the domain . The paper looks at differences between weak and strong removability,the domain dependence of strong removability, and when removabilityis preserved under unions. In particular, a domain and a setE that is weakly removable for all H^p, but not strongly removablefor any H^p(\E), 0 < p < , are found. It is easy to show that if E is weakly removable for H^p(\E)and q > p, then E is also weakly removable for H^q(\E). Itis shown that the corresponding implication for strong removabilityholds if and only if q/p is an integer. Finally, the theory of Hardy space capacities is extended, anda comparison is made with the similar situation for weightedBergman spaces.     

An Access Theorem for Continuous Functions

Let f be a continuous function on an open subset of R² suchthat for every x there exists a continuous map : [–1,1] with (0) = x and f increasing on [–1, 1]. Thenfor every there exists a continuous map : [0, 1) suchthat (0) = y, f is increasing on [0; 1), and for every compactsubset K of , max{t : (t) K} < 1. This result gives an answerto a question posed by M. Ortel. Furthermore, an example showsthat this result is not valid in higher dimensions.     

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).     

Recurrence, Dimension and Entropy

Let (_A, T) be a topologically mixing subshift of finite typeon an alphabet consisting of m symbols and let :_A R^d be a continuousfunction. Denote by (x) the ergodic limit when the limit exists. Possible ergodic limits arejust mean values dµ for all T-invariant measures. Forany possible ergodic limit , the following variational formulais proved: where h_µ denotes the entropy of µ and h_top denotestopological entropy. It is also proved that unless all pointshave the same ergodic limit, then the set of points whose ergodiclimit does not exist has the same topological entropy as thewhole space _A     

The Continuous Postage Stamp Problem

For a real set A consider the semigroup S(A), additively generatedby A; that is, the set of all real numbers representable asa (finite) sum of elements of A. If A (0, 1) is open and non-empty,then S(A) is easily seen to contain all sufficiently large realnumbers, and we let G(A): = sup{u R: u S(A)}. Thus G(A) isthe smallest number with the property that any u > G(A) isrepresentable as indicated above. We show that if the measure of A is large, then G(A) is small;more precisely, writing for brevity : = mes A, we have Indeed, the first and the last of these three estimates arethe best possible, attained for A = (1–, 1) and A = (1–,1)\{2(1–)}, respectively; the second is close to the bestpossible and can be improved by {1/}1/ {1/} at most. The problem studied is a continuous analogue of the linear Diophantineproblem of Frobenius (in its extremal settings due to Erdösand Graham), also known as the ‘postage stamp problem’or the ‘coin exchange problem’.     

Approximately Multiplicative Functionals on Algebras of Smooth Functions

Let be a bounded linear functional on A, where A is a commutativeBanach algebra, then the bilinear functional is defined as (a,b)= (ab)- (a) (b) for each a and b in A. If the norm of is small then is approximatelymultiplicative, and it is of interest whether or not |||| being small implies that is nearto a multiplicative functional. If this property holds for acommutative Banach algebra then A is an AMNM algebra (approximatelymultiplicative functionals are near multiplicative functionals).The main result of the paper shows that C^N [0,1]^M (the complexvalued functions defined on [0,1]^M with all Nth order partialderivatives continuous) is AMNM. It is also shown that a similarproof can be applied to certain Lipschitz algebras.     

Optimal Cardinals for Metrizable Barrelled Spaces

We seek the smallest or largest cardinals for which certainbasic results hold, as did Mazur when he proved that c is thesmallest infinite-dimensionality for a Fréchet space.As with Mazur, we make no axiomatic assumptions outside theusual ZFC model. We discover three instances in which the optimalcardinal is the dominating number and three in which it isthe bounding number b, apparently giving the first locally convexspace characterizations of these venerable and easily describedcardinals. Here are two samples: it is known that for any non-normablemetrizable locally convex space E, the minimal size _b(E) fora fundamental system of bounded sets must satisfy _{1 b}(E) c;we prove that _b(E) = . Again, it is known that if E is a non-normablemetrizable barrelled space of minimal dimension, then ₁ dim(E) c; we prove that dim(E) = b. The most important individualresult is the reconstruction of Tweddle's space without useof the Continuum Hypothesis (₁ = c). The reconstruction is vitalin the characterizations of b and in subsequent papers answeringopen questions about countable enlargements.     

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.     

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.