Interpolation sets and the Bohr topology of locally compact groups

Rosenthal's theorem describing those Banach spaces containing no copy of ?¹ is extended to topological groups replacing ?¹-basis by interpolation sets in the sense of Hartman and Ryll-Nardzewsky (Colloq. Math. 12 (1964) 23-39). This extension provides a characterization of those locally compact groups containing no interpolation sets and of those locally compact groups which respect compactness, i.e, such that every Bohr compact subset is compact. The approach followed in this paper sheds some light on other questions related to the duality theory of non-Abelian locally compact groups.     

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.     

Remarks on star countable discrete closed spaces

In this paper, we prove the following statements:

1. There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable.
2. Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace.
3. Assuming $2^{\aleph _0 } = 2^{\aleph _1 } $ , there exists a normal absolutely star countable discrete closed space having a regular-closed subspace which is not star countable.

     

Low-dimensional topology and ordering groups

This expository paper explores the interaction of group ordering with topological questions, especially in dimensions 2 and 3. Among the topics considered are surfaces, braid groups, 3-manifolds and their structures such as foliations and mappings between them. A final section explores currently ongoing research regarding spaces of homeomorphisms and their orderability properties. This is not meant to be a comprehensive survey, but rather just a taste of the rich relationship between topology and the theory of ordered groups.     

Fuzzy topology on fuzzy sets: Product fuzzy topology and fuzzy topological groups

The notion of product fuzzy topology in the case of fuzzy topology on fuzzy sets is introduced and the product invariance of fuzzy Hausdorffness, compactness, connectedness are examined. The product fuzzy topology is used to define fuzzy group topology on a fuzzy subgroup of a group G and some properties of fuzzy topological groups are obtained.     

Remarks on Lipschitz properties of matrix groups actions

It is proved that a large class of matrix group actions, including joint similarity and congruence-like actions, as well as actions of the type of matrix equivalence, have local Lipschitz property. Under additional hypotheses, global Lipschitz property is proved. These results are specialized and applied to obtain local Lipschitz property of canonical bases of matrices that are selfadjoint in an indefinite inner product. Real, complex, and quaternionic matrices are considered.     

Topologies on groups determined by discrete subsets

Let G be an infinite group. Given a filter F on G, let T[F] denote the largest left invariant topology on G in which F converges to the identity. In this paper, we study the topology T[F] in case when F contains the Fréchet filter and there is such that all the subsets xM(x), where x∈G, are pairwise disjoint. We show that T[F] possesses interesting extremal properties. We consider also the question whether T[F] can be a group topology.     

Convolution of discrete measures on linear groups

Let p,q∈R such that 1<p<2 and . Define

(∗)     

Convolution of discrete measures on linear groups

Let p,q∈R such that 1<p<2 and . Define

(∗)     

Infinitely divisible probabilities on discrete linear groups

We investigate the structure of infinitely divisible probability measures on a discrete linear group. It is shown that for any such measure there is an infinitely divisible elementz in the centralizer of the support of the measure, such that the translate of the measure byz is embeddable over the subgroup generated by the support of the measure. Examples are given to show that this reult is best possible.     

Characters andK-theory of discrete groups

Erratum

Characters andK-theory of discrete groups