Low-dimensional unitary representations of

We characterize all simple unitarizable representations of the braid group on complex vector spaces of dimension . In particular, we prove that if and denote the two generating twists of , then a simple representation (for ) is unitarizable if and only if the eigenvalues of are distinct, satisfy and 0$"> for , where the are functions of the eigenvalues, explicitly described in this paper.

     

Planar algebras and the Ocneanu-Szymanski theorem

We give a very simple `planar algebra' proof of the part of the Ocneanu-Szymanski theorem which asserts that for a finite index, depth two, irreducible -subfactor , the relative commutants and admit mutually dual Kac algebra structures. In the hyperfinite case, the same techniques also prove the other part, which asserts that acts on with invariants .

     

A note on -bases of a regular affine domain extension

Let be a tower of commutative rings where is a regular affine domain over an algebraically closed field of prime characteristic and is a regular domain. Suppose has a -basis over and . For a subset of whose elements satisfy a certain condition on linear independence, let be a set of maximal ideals of such that is a -basis of over . We shall characterize this set in a geometrical aspect.

     

Minimal rank and reflexivity of operator spaces

Let be an -dimensional space of linear operators between the linear spaces and over an algebraically closed field . Improving results of Larson, Ding, and Li and Pan we show the following.

Theorem. Let be a basis of . Assume that every nonzero operator in has rank larger than . Then a linear operator belongs to if and only if for every , is a linear combination of .

     

Banach-Stone theorem for Banach lattice valued continuous functions

Let and be compact Hausdorff spaces, be a Banach lattice and be an AM space with unit. Let be a Riesz isomorphism such that if and only if for each . We prove that is homeomorphic to and is Riesz isomorphic to . This generalizes some known results.

     

On conjugation invariants in the dual Steenrod algebra

We investigate the canonical conjugation, , of the mod dual Steenrod algebra, , with a view to determining the subspace, , of elements invariant under . We give bounds on the dimension of this subspace for each degree and show that, after inverting , it becomes polynomial on a natural set of generators. Finally we note that, without inverting , is far from being polynomial.

     

Eigenvalue fields of hyperbolic orbifolds

In this paper, we prove that if is a non-elementary subgroup of , with , then the eigenvalue field of has infinite degree over .

     

Growth of solutions of second order linear differential equations

We consider the differential equation , where and are entire functions. Provided and as outside a set of finite logarithmic measure, we prove that all nonconstant solutions of this equation are of infinite order.

     

Fourier asymptotics of Cantor type measures at infinity

Let be an integer and let . In this note we prove that for all ; if is odd and if is even This improves a classical result of Wiener and Wintner. We also give a necessary and sufficient condition for the product to approach zero at infinity.

     

-isomorphisms, Jordan isomorphisms, and numerical range preserving maps

Let or , where is the algebra of a bounded linear operator acting on the Hilbert space , and is the set of self-adjoint operators in . Denote the numerical range of by It is shown that a surjective map satisfies

if and only if there is a unitary operator such that has the form

where is the transpose of with respect to a fixed orthonormal basis. In other words, the map or is a -isomorphism on and a Jordan isomorphism on . Moreover, if has finite dimension, then the surjective assumption on can be removed.

     

Local existence of -sets, projective tensor products, and Arens regularity for

Theorem. If are perfect compact subsets of the locally compact metrizable abelian group, then there are pairwise disjoint perfect subsets such that (i) is either a Kronecker set or (ii) for some , is a translate of a -set all of whose elements have order , and (iii) is isomorphic to the projective tensor product .

This extends what was previously known for groups such as or for the case to the general locally compact abelian group. Old results concerning the local existence of Kronecker and -sets are improved.

     

A note on the imbedding theorem of Browder and Ton

The imbedding theorem of Browder and Ton states that for any real separable Banach space there exist a real separable Hilbert space and a compact linear injection such that is dense in We shall give a short and elementary new proof to this result. We also briefly discuss the corresponding result without the completeness assumption.

     

Operators with bounded conjugation orbits

For a bounded invertible operator on a complex Banach space let be the set of operators in for which Suppose that and is in A bound is given on in terms of the spectral radius of the commutator. Replacing the condition in by the weaker condition as for every 0$">, an extension of the Deddens-Stampfli-Williams results on the commutant of is given.

     

Functions -orthogonal with respect to their own zeros

In 1939, G. H. Hardy proved that, under certain conditions, the only functions satisfying

where the are the zeros of , are the Bessel functions. We replace the above integral by the Jackson -integral and give the -analogue of Hardy's result.

     

On finite groups admitting a special noncoprime action

An important result of Turull (1984) is the following:

Let be a finite solvable group, and . Then , where denotes the Fitting height and denotes the composition length.

The purpose of this work is to give a treatment of the minimal configuration in this framework with additional conditions, yet without the coprimeness condition.

     

Row and column finite matrices

Consider the ring of all column finite matrices over a ring . We prove that each such matrix is conjugate to a row and column finite matrix if and only if is right Noetherian and is countable. We then demonstrate that one can perform this conjugation on countably many matrices simultaneously. Some applications and limitations are given.

     

On local irreducibility of the spectrum

Let be the algebra of complex matrices, and for denote by and the spectrum and spectral radius of respectively. Let be a domain in containing 0, and let be a holomorphic map. We prove: (1) if for , then for ; (2) if for , then again for . Both results are special cases of theorems expressing the irreducibility of the spectrum near .

     

Pure Picard-Vessiot extensions with generic properties

Given a connected linear algebraic group over an algebraically closed field of characteristic 0, we construct a pure Picard-Vessiot extension for , namely, a Picard-Vessiot extension , with differential Galois group , such that and are purely differentially transcendental over . The differential field is the quotient field of a -stable proper differential subring with the property that if is any differential field with field of constants and is a Picard-Vessiot extension with differential Galois group a connected subgroup of , then there is a differential homomorphism such that is generated over as a differential field by .

     

A numerical condition for a deformation of a Gorenstein surface singularity to admit a simultaneous log-canonical model

Let be a deformation of a normal Gorenstein surface singularity over the complex number field . We assume that is a neighborhood of the origin of . Then we prove that admits a simultaneous log-canonical model if and only if an invariant of each fiber is constant.