Topological dynamics on moduli spaces II

Let be an orientable genus 0$"> surface with boundary . Let be the mapping class group of fixing . The group acts on the space of -gauge equivalence classes of flat -connections on with fixed holonomy on . We study the topological dynamics of the -action and give conditions for the individual -orbits to be dense in .

     

Hyponormality of trigonometric Toeplitz operators

In this paper we establish a tractable and explicit criterion for the hyponormality of arbitrary trigonometric Toeplitz operators, i.e., Toeplitz operators with trigonometric polynomial symbols . Our criterion involves the zeros of an analytic polynomial induced by the Fourier coefficients of . Moreover the rank of the selfcommutator of is computed from the number of zeros of in the open unit disk and in counting multiplicity.

     

Local subgroups and the stable category

If is a finite group and is an algebraically closed field of characteristic 0$">, then this paper uses the local subgroup structure of to define a category that is equivalent to the stable category of all left -modules modulo projectives. A subcategory of equivalent to the stable category of finitely generated -modules is also identified. The definition of depends largely but not exclusively upon local data; one condition on the objects involves compatibility with respect to conjugations by arbitrary group elements rather than just elements of -local subgroups.

     

Sets of uniqueness for spherically convergent multiple trigonometric series

A subset of the -dimensional torus is called a set of uniqueness, or -set, if every multiple trigonometric series spherically converging to outside vanishes identically. We show that all countable sets are -sets and also that sets are -sets for every . In particular, , where is the Cantor set, is an set and hence a -set. We will say that is a -set if every multiple trigonometric series spherically Abel summable to outside and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for sets. In addition, every -set has measure , and a countable union of closed -sets is a -set.

     

On the Glauberman and Watanabe correspondences for blocks of finite -solvable groups

If is a finite -solvable group for some prime , a solvable subgroup of the automorphism group of of order prime to such that stabilises a -block of and acts trivially on a defect group of , then there is a Morita equivalence between the block and its Watanabe correspondent of , given by a bimodule with vertex and an endo-permutation module as source, which on the character level induces the Glauberman correspondence (and which is an isotypy by Watanabe's results).

     

Cotensor products of modules

Let be a coalgebra over a field and its dual algebra. The category of -comodules is equivalent to a category of -modules. We use this to interpret the cotensor product of two comodules in terms of the appropriate Hochschild cohomology of the -bimodule , when is finite-dimensional, profinite, graded or differential-graded. The main applications are to Galois cohomology, comodules over the Steenrod algebra, and the homology of induced fibrations.

     

Vertices for characters of -solvable groups

Suppose that is a finite -solvable group. We associate to every irreducible complex character of a canonical pair , where is a -subgroup of and , uniquely determined by up to -conjugacy. This pair behaves as a Green vertex and partitions into ``families" of characters. Using the pair , we give a canonical choice of a certain -radical subgroup of and a character associated to which was predicted by some conjecture of G. R. Robinson.

     

Trees and branches in Banach spaces

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree of a certain type on a space is presumed to have a branch with some property. It is shown that then can be embedded into a space with an FDD so that all normalized sequences in which are almost a skipped blocking of have that property. As an application of our work we prove that if is a separable reflexive Banach space and for some and every weakly null tree on the sphere of has a branch -equivalent to the unit vector basis of , then for all 0$">, there exists a subspace of having finite codimension which embeds into the sum of finite dimensional spaces.

     

On a class of jointly hyponormal Toeplitz operators

We characterize when a pair of Toeplitz operators is jointly hyponormal under various assumptions--for example, is analytic or is a trigonometric polynomial or is analytic. A typical characterization states that is jointly hyponormal if and only if an algebraic relation of and holds and the single Toeplitz operator is hyponormal, where is a combination of and . More general results for an -tuple of Toeplitz operators are also obtained.