Bounded and completely bounded local derivations from certain commutative semisimple Banach algebras

We show that for a locally compact group , every completely bounded local derivation from the Fourier algebra into a symmetric operator -module or the operator dual of an essential -bimodule is a derivation. Moreover, for amenable we show that the result is true for all operator -bimodules. In particular, we derive a new proof to a result of N. Spronk that is always operator weakly amenable.

     

Transference for amenable actions

Suppose acts amenably on a measure space with quasi-invariant -finite measure . Let be an isometric representation of on and a finite Radon measure on . We show that the operator has -operator norm not exceeding the -operator norm of the convolution operator defined by . We shall also prove an analogous result for the maximal function associated to a countable family of Radon measures .

     

Detecting the index of a subgroup in the subgroup lattice

A theorem by Zacher and Rips states that the finiteness of the index of a subgroup can be described in terms of purely lattice-theoretic concepts. On the other hand, it is clear that if  is a group and is a subgroup of finite index of , the index cannot be recognized in the lattice of all subgroups of , as for instance all groups of prime order have isomorphic subgroup lattices. The aim of this paper is to give a lattice-theoretic characterization of the number of prime factors (with multiplicity) of .

     

Invariant ideals of abelian group algebras under the multiplicative action of a field. I

Let be a division ring and let be a finite-dimensional -vector space, viewed multiplicatively. If is the multiplicative group of , then acts on and hence on any group algebra . Our goal is to completely describe the semiprime -stable ideals of . As it turns out, this result follows fairly easily from the corresponding results for the field of rational numbers (due to Brookes and Evans) and for infinite locally-finite fields. Part I of this work is concerned with the latter situation, while Part II deals with arbitrary division rings.

     

A variant of the Reynolds operator

Let be a linearly reductive group over a field , and let be a -algebra with a rational action of . Given rational --modules and , we define for the induced -action on Hom a generalized Reynolds operator, which exists even if the action on Hom is not rational. Given an -module homomorphism , it produces, in a natural way, an -module homomorphism which is -equivariant. We use this generalized Reynolds operator to study properties of rational - modules. In particular, we prove that if is invariantly generated (i.e. ), then is a projective (resp. flat) -module provided that is a projective (resp. flat) -module. We also give a criterion whether an -projective (or -flat) rational --module is extended from an -module.

     

The Hausmann-Weinberger 4-manifold invariant of abelian groups

The Hausmann-Weinberger invariant of a group is the minimal Euler characteristic of a closed orientable 4-manifold with fundamental group . We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.

     

Blocks of central -group extensions

Let and be finite groups that have a common central -subgroup for a prime number , and let and respectively be -blocks of and induced by -blocks and respectively of and , both of which have the same defect group. We prove that if and are Morita equivalent via a certain special -bimodule, then such a Morita equivalence lifts to a Morita equivalence between and .

     

Invariant ideals of abelian group algebras under the multiplicative action of a field. II

Let be a division ring and let be a finite-dimensional right -vector space, viewed multiplicatively. If is the multiplicative group of , then acts on and hence on any group algebra . In this paper, we completely describe the semiprime -stable ideals of , and conclude that these ideals satisfy the ascending chain condition. As it turns out, this result follows fairly easily from the corresponding results for the field of rational numbers (due to Brookes and Evans) and for infinite locally-finite fields (handled in Part I).

     

A note on principal parts on projective space and linear representations

Let be a closed subgroup of a linear algebraic group defined over a field of characteristic zero. There is an equivalence of categories between the category of linear finite-dimensional representations of , and the category of finite rank -homogeneous vector bundles on . In this paper we will study this correspondence for the sheaves of principal parts on projective space, and we describe the representation corresponding to the principal parts of a line bundle on projective space.

     

Extreme points of the unit ball of the Fourier-Stieltjes algebra

Let be a locally compact group. Among other things, we proved in this paper that for an IN-group , the extreme points of the unit ball of the Fourier-Stieltjes algebra are not in the Fourier algebra if and only if is non-compact, or equivalently, there is no irreducible representation of which is quasi-equivalent to a subrepresentation of the left regular representation of if and only if is non-compact. This result is a non-commutative version of the following well known result: For any locally compact group , the extreme points of the unit ball of the measure algebra are not in the group algebra if and only if is non-discrete. On the other hand, we also showed that if has the RNP, then there are extreme points of the unit ball of that are in . Since it is well known there are non-compact locally compact group for which has the RNP, there exist non-compact locally compact groups where extreme points of the unit ball of can be in . This shows that the condition be an IN-group cannot be entirely removed.