On abelian, triangularizable, total reduction algebras

Suppose that is a complex Hilbert space and that () denotes the bounded linear operators on .In this paper we show that if () is an abelian algebrawith elements that admit representations under which they havea spanning set of eigenvectors, and if has the property that given any bounded representation: () of ona Hilbert space , every invariant subspace of () is topologically complemented by an invariantsubspace of (), then is similar to an abelian C^*-algebra. From thisit follows that any amenable Banach algebra of triangular operatorson a separable Hilbert space is similar to an abelian C^*-algebra.     

One-sided ideals and right cancellation in the second dual of the group algebra and similar algebras

The following results are proved for a non-compact, locallycompact group G: the dimension of every non-trivial right idealin L¹(G)** (equipped with the first Arens product) is at least, where (G) is the minimalnumber of compact sets required to cover G; there exist left ideals in L¹(G)** and in LUC(G)* with trivialintersections, and the linear span of right-cancellable elementsis weak*-dense in the annihilator of C₀(G) in LUC(G)* and inthe annihilator of (theL-functions that vanish at infinity) in L(G)*. The same resultsare proved for weighted algebras when the weight function isdiagonally bounded.     

On products of quadratic twists and ranks of elliptic curves over large fields

In this paper, we give examples of elliptic curves E/K overa number field K satisfying the property that there exist P₁,P₂ K[t] such that the twists and are of positiverank over K(t). As a consequence of this result on twists, weshow that for those elliptic curves E/K, and for each , the rank of E over the fixed field (K^ab) under is infinite, where K^ab is the maximal abelian extension ofK.     

Ultrarapidly decreasing ultradifferentiable functions, Wigner distributions and density matrices

Spaces , , of ultradecreasing ultradifferentiable (or forshort, ultra-) functions, depending on a weight e^(x), are introducedin the context of quantum statistics. The corresponding coefficientspaces in the Fock basis are identified, and it is shown thatthe Hermite expansion is a tame isomorphism between these spaces.These results are used to link decay properties of density matricesto corresponding properties of the Wigner distribution.     

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.     

The integral Novikov conjectures for linear groups containing torsion elements

In this paper, we show that for any global field k, the generalizedintegral Novikov conjecture in both K- and L-theories holdsfor every finitely generated subgroup of GL(n, k). This impliesthat the conjecture holds for every finitely generated subgroupof , where is the algebraic closure of . We also show that for every linear algebraicgroup defined over k, every S-arithmetic subgroup satisfiesthis generalized integral Novikov conjecture. We note that theintegral Novikov conjecture implies the stable Borel conjecture,in particular, the stable Borel conjecture holds for all theabove torsion-free groups. Most of these subgroups are not discretesubgroups of Lie groups with finitely many connected components,and some of them are not finitely generated. When the fieldk is a function field such as , and the k-rank of is positive, many of these S-arithmeticsubgroups such as donot admit cofinite universal spaces for proper actions. Received February 15, 2007.     

Equations Defining Symmetric Varieties and Affine Grassmannians

Suppose be a simpleinvolution of a semisimple algebraic group and suppose H is the subgroup of G of pointsfixed by . If the restrictedroot system is of type or and G is simplyconnected, or if the restricted root system is of type and G is of adjoint type, then we describe astandard monomial theory and the equations for the coordinatering using the standardmonomial theory and the Plücker relations of an appropriate(maybe infinite-dimensional) Grassmann variety.     

Radial Components, Prehomogeneous Vector Spaces, and Rational Cherednik Algebras

Let be a finite-dimensionalrepresentation of a connected reductive complex Lie group . Denote by G the derived subgroup of and assume that the categorical quotient is one dimensional, i.e. for a nonconstant polynomial f. In this situationthere exists a homomorphism , the radial component map, where is the first Weyl algebra. We show that theimage of is isomorphicto the spherical subalgebra of a rational Cherednik algebrawhose multiplicity function is defined by the roots of the Bernstein–Satopolynomial of f. In the case where is also multiplicity free we describe the kernel of and prove a Howe duality result between representationsof G occurring in andlowest-weight modules over the Lie algebra generated by f andthe "dual" differential operator ; this extends results of H. Rubenthaler obtained when is a parabolic prehomogeneous vector space.If satisfies a Capelli-typecondition, some applications are given to holonomic and equivariantD-modules on V. These applications are related to results provedby Muro [34, 37, 38] or Nang [40, 42, 44] in special cases ofthe representation .     

Zeta Functions for G2 and Their Zeros

The exceptional group G₂ has two maximal parabolic subgroups corresponding to theso-called long root and short root. In this paper, the secondnamed author introduces two zeta functions associated with and respectively, and the first named author proves that these zetassatisfy the Riemann hypothesis.     

Structure of bounded topological-sequence-entropy minimal systems

In this article we prove that a minimal topological dynamicalsystem (X, T) with bounded topological sequence entropy hasthe following structure. Here is the maximal equicontinuous factor of (X, T), ^' and^' are proximal extensions and ^' is a finite-to-one equicontinuousextension. In order to prove this result we consider sequenceentropy tuples and give their complete relation with regionallyproximal tuples.     

Spectral properties of banded Toeplitz matrices * By Albrecht Bottcher and Sergei M. Grudsky: * 411 pp., US$ 95.00, ISBN 0-89871-599-7 * (Society for Industrial and Applied Mathematics, Philadelphia, 2005)

Recall that an infinite Toeplitz matrix, , say, is one for which the values t_jk dependonly on j – k, so we may write t_jk = a_{j – k}; thusthe entries are constant on diagonals sloping down from topleft to bottom right:

Sucha matrix is said to be banded if there are only finite manynon-zero a_j, say . The propertiesof such a matrix, when defining a linear operator T on a sequencespace, are often closely related to the properties of the Laurentpolynomial

for zlying on the unit circle in the complex plane. For example,suppose that the matrix acts on the most familiar sequence space,the Hilbert space ²; then it has been known for many years thatthe operator norm of T is the maximum     

Operations on the A-theoretic nil-terms

For a space X, we define Frobenius and Verschiebung operationson the nil-terms inthe algebraic K-theory of spaces, in three different ways. Twoapplications are included. Firstly, we show that the homotopygroups of are eithertrivial or not finitely generated as abelian groups. Secondly,the Verschiebung operation defines a -module structure on the homotopy groups of , with the multiplicative monoid. We also give a calculation of the homotopy type of the nil-terms after p-completion foran odd prime p and their homotopy groups as -modules up to dimension 4p – 7. We obtainnon-trivial groups only in dimension 2p – 2, where itis finitely generated as a -module, and in dimension 2p – 1, where it is not finitelygenerated as a -module. Received April 12, 2007.     

The symplectic ideal and a double centraliser theorem

We interpret a result of Oehms as a statement about the symplecticideal. We use this result to prove a double centraliser theoremfor the symplectic group acting on , where V is the natural module for the symplectic group.This result was obtained in characteristic zero by Weyl. Furthermore,we use this to extend to arbitrary connected reductive groupsG with simply connected derived group the earlier result ofthe author that the algebra K[G] of infinitesimal invariantsin the algebra of regular functions on G is a unique factorisationdomain.     

UNEXPECTED SUBSPACES OF TENSOR PRODUCTS

We describe complemented copies of ₂ both in C(K₁) C(K₂) when at least one of the compact spaces K_iis not scattered and in L₁(µ₁)L₁(µ₂) when at least one of the measures is not atomic.The corresponding local construction gives uniformly complementedcopies of the in c₀ c₀. We continue the study of c₀ c₀ showing that it contains a complementedcopy of Stegall's space and proving that (c₀ c₀)' is isomorphicto , together with other results. In the last section we use Hardy spaces to find an isomorphiccopy of L_p in the space of compact operators from L_q to L_r,where 1 < p, q, r < and 1/r = 1/p + 1/q.     

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.     

Interpolation of Vector-Valued Real Analytic Functions

Let R^d be an open domain. The sequentially complete DF-spacesE are characterized such that for each (some) discrete sequence(z_n) , a sequence of natural numbers (k_n) and any family the infinite system of equations has an E-valued real analytic solution f.     

Traces and extensions of matrix-weighted Besov spaces

Let V be a matrix weight on ⁿ⁺¹ and let W be a matrix weighton ⁿ, satisfying, for example, the matrix A_p condition. Definethe trace, or restriction, operator Tr by Tr (f)(x^')=f(x^', 0),where x^'n and f is a function on ⁿ⁺¹. If –1/p>n (1/p–1)₊+(β–n)/p,where β is the doubling exponent of W, then the trace operatoris bounded from into (matrix-weighted Besov spaces) if and only ifthe weights V and W uniformly satisfy an estimate controllingthe average of on anydyadic cube I ⁿ by the average of on Q(I)=Ix[0, (I)], for all . If V and W satisfy the converse inequality, then there existsa continuous linear map .If both inequalities hold, then Tr Ext is the identity on .     

Prime paucity for sums of two squares

Almost all solutions of the diophantine equations , with all x_j prime numbers, are diagonal ones.This is established here in quantitative form.     

The Singular Homology of the Hawaiian Earring

The singular homology groups of compact CW-complexes are finitelygenerated, but the groups of compact metric spaces in generalare very easy to generate infinitely and our understanding ofthese groups is far from complete even for the following compactsubset of the plane, called the Hawaiian earring: Griffiths [11] gave a presentation of the fundamental groupof H and the proof was completed by Morgan and Morrison [15].The same group is presented as the free -product of integers Z in [4, Appendix]. Hence the firstintegral singular homology group H₁(H) is the abelianizationof the group . These results have been generalized to non-metrizable counterparts H_I of H(see Section 3). In Section 2 we prove that H₁(X) is torsion-free and H_i(X) =0 for each one-dimensional normal space X and for each i 2.The result for i 2 is a slight generalization of [2, Theorem5]. In Section 3 we provide an explicit presentation of H₁(H)and also H₁(H_I) by using results of [4]. Throughout this paper, a continuum means a compact connectedmetric space and all maps are assumed to be continuous. Allhomology groups have the integers Z as the coefficients. Thebouquet with n circles is denoted by B_n. The base point (0, 0) of B_n is denoted by o forsimplicity.     

Link Polynomials of Higher Order

In this paper, we study certain polynomial invariants of links(singular or non-singular) that are related to the Homfly polynomialand Vassiliev's invariants. The Homfly polynomial H_L [3] (alsoknown as the Flypmoth polynomial) satisfies the well-known skeinrelation The Vassiliev invariants [1, 2] (of order 1) satisfy the relations and The invariants that we study satisfy the skein relations