Malliavin's theorem for weak synthesis on nonabelian groups

Malliavin's celebrated theorem on the failure of spectral synthesis for the Fourier algebra A(G) on nondiscrete abelian groups was strengthened to give failure of weak synthesis by Parthasarathy and Varma. We extend this to nonabelian groups by proving that weak synthesis holds for A(G) if and only if G is discrete. We give the injection theorem and the inverse projection theorem for weak X-spectral synthesis, as well as a condition for the union of two weak X-spectral sets to be weak X-spectral for an A(G)-submodule X of VN(G). Relations between weak X-synthesis in A(G) and A(G×G) and the Varopoulos algebra V(G) are explored. The concept of operator synthesis was introduced by Arveson. We extend several recent investigations on operator synthesis by defining and studying, for a V^∞(G)-submodule M of B(L²(G)), sets of weak M-operator synthesis. Relations between X-Ditkin sets and M-operator Ditkin sets and between weak X-spectral synthesis and weak M-operator synthesis are explored.     

The weak isovariant Borsuk-Ulam theorem for compact Lie groups

In this paper we shall deal with a weak version of the Borsuk-Ulam theorem for G-isovariant maps, which we call the weak isovariant Borsuk-Ulam theorem. One of the results is that the weak isovariant Borsuk-Ulam theorem in linear G-spheres holds for an arbitrary compact Lie group G. On the contrary the weak isovariant Borsuk-Ulam theorem in semilinear G-(homology) spheres holds if and only if G is solvable. Received: 2 April 2002     

Remarks on free Bieberbach groups

Recent results on the Gupta representation of the unique stem cover of a free Bieberbach group lead to explicite determination of its Schur multiplicator. The aim of this note is to estimate the free-abelian rank of the Schur multiplicator, using only the 8-term Eckmann-Hilton-Stammbach exact homology sequence induced by the free Bieberbach group modulo its nontrivial center. The note ends with a discussion about the structure of torsion-free Bieberbach groups constructed by S.Rosset.     

A structure theorem for groups of weak characteristic p-type

We prove a structure theorem for groups of weak characteristic p-type. We use this result to explain how to identify an "end-node" of a diagram for a group.     

A nonstandard density theorem for weak topologies on Banach and Bochner spaces

We prove a nonstandard density result. It asserts that if a particular formula is true for functions in a set K of linear continuous functions between Banach spaces E and D, then it remains valid for functions that are limits, in the uniform convergence topology on a given class ?? of subsets of E, of nets of vectors in K. We then apply this result to various class ?? and setsK in the context of E‐valued Bochner integrable functions defined on a finite measure space.     

Fourier multiplier theorem for atomic Hardy spaces on unbounded Vilenkin groups

We characterize atomic Hardy spaces on unbounded locally compact Vilenkin groups by means of a modified maximal function. The obtained Fourier multiplier theorem is more general than the corresponding results due to Kitada, Onneweer and Quek, Daly and Phillips that were proved under the boundedness assumption on the underlying group.     

On the Charshiladze-Lozinski theorem for compact topological groups and homogeneous spaces

This paper is concerned with an extension of the Charshiladze-Lozinski theorem to compact (not necessarily abelian) topological groups G and symmetric compact homogeneous spaces G/H. The proof is based on a generalized Marcinkiewicz — Berman formula. As an application, some divergence theorems for expansions of continuous resp. integrable complex — valued functions on Euclidean spheres and projective spaces in series of polynomial functions on these spaces are established.     

The Levin-Golovin theorem for sobolev spaces

We obtain sufficient conditions for the basis property of the family of exponentials

弱模余代数的结构定理

设H为弱Hopf代数,C为弱右H-模余代数,令C=C/C·ker L.利用Smash余积来研究弱模余代数上的结构定理,并给出了C与C×H作为余代数同构的条件.     

The Baire category theorem in products of linear spaces and topological groups

A space is a Baire space if the intersection of countably many dense open sets is dense. We show that if X is a non-separable completely metrizable linear space (pathconnected abelian topological group) then X contains two linear subspaces (subgroups) E and F such that both E and F are Baire but E×F is not. If X is a completely metrizable linear space of weight ℵ₁ then X is the direct sum E⊕F of two linear subspaces E and F such that both E and F are Baire but E×F is not.     

The Auslander conjecture for NIL-affine crystallographic groups

Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups in Aff(N)=NAut(N) acting properly discontinuously and cocompactly on N. This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions 1n5 the generalized Auslander conjecture holds, i.e., that such subgroups are virtually polycyclic.     

The structure theorem for weak module coalgebras

Let H be a weak Hopf algebra, let C be a weak right H-module coalgebra, and let $ \bar C = {C \mathord{\left/ {\vphantom {C C}} \right. \kern-0em} C} \cdot Ker \sqcap ^L $ . We prove a structure theorem for weak module coalgebras, namely, C is isomorphic as a weak right H-module coalgebra to a weak smash coproduct $ \bar C $ × H defined on a k-space $$ \{ \Sigma c_{(0)} \otimes h_2 \varepsilon (c_{( - 1)} h_1 )|c \in C,h \in H\} $$ if there exists a weak right H-module coalgebra map ?: C → H.