Pseudocompact topological group refinements of maximal weight

It is known that a compact metrizable group admits no proper pseudocompact topological group refinement. The authors show, in contrast, that every (Hausdorff) pseudocompact Abelian group of uncountable weight , satisfying any of the following conditions, admits a pseudocompact group refinement of maximal weight (that is, of weight ):

(i)

is compact;

(ii)

is torsion-free with ;

(iii)

[GCH] is torsion-free.

Remark. (i) answers a question posed by Comfort and Remus [Math. Zeit- schrift 215 (1994), 337-346].

     

Recovery of Band-Limited Functions on Locally Compact Abelian Groups from Irregular Samples

Using the techniques of approximation and factorization of convolution operators we study the problem of irregular sampling of band-limited functions on a locally compact Abelian group G. The results of this paper relate to earlier work by Feichtinger and Gröchenig in a similar way as Kluvánek's work published in 1969 relates to the classical Shannon Sampling Theorem. Generally speaking we claim that reconstruction is possible as long as there is sufficient high sampling density. Moreover, the iterative reconstruction algorithms apply simultaneously to families of Banach spaces.     

Transversality on locally pseudocompact groups

Two non-discrete Hausdorff group topologies \tau and \delta on a group G are called transversal if the least upper bound \tau \vee \delta of \tau and \delta is the discrete topology. In this paper, we discuss the existence of transversal group topologies on locally pseudocompact, locally precompact, or locally compact groups. We prove that each locally pseudocompact, connected topological group satisfies central subgroup paradigm, which gives an affrmative answer to a problem posed by Dikranjan, Tkachenko, and Yaschenko [Topology Appl., 2006, 153:3338-3354]. For a compact normal subgroup K of a locally compact totally disconnected group G, if G admits a transversal group topology, then G/K admits a transversal group topology, which gives a partial answer again to a problem posed by Dikranjan, Tkachenko, and Yaschenko in 2006. Moreover, we characterize some classes of locally compact groups that admit transversal group topologies.     

Sixteen-Dimensional Locally Compact Translation Planes with Large Automorphism Groups having no Fixed Points

The 16-dimensional compact projective planes whose automorphism group contains a closed connected subgroup fixing a line, but no point and having dimension at least 35 are determined. It is shown that these planes all belong to three families of planes determined by H. Löwe and the author, and hence are explicitly known. A major stepping stone to this goal is a result by H. Salzmann according to which every such plane is a translation plane.     

Contractive Automorphisms of Locally Compact Groups and the Concentration Function Problem

Let G be a noncompact locally compact group. We show that a necessary and sufficient condition in order that G support an adapted probability measure whose concentration functions fail converge to zero is that G be the semidirect product , where is an automorphism of N contractive modulo a compact subgroup. Any adapted a probability measure whose concentration functions fail to converge to zero has the form =v×₁ where v is a probability measure on N. If G is unimodular then the concentration functions of an adapted probability measure fail to converge to zero if and only if is supported on a coset of a compact normal subgroup.     

最大Abel商群为局部循环群的可解群

从自同构群的角度出发,给出了一些具有有限性条件的、最大Abel商群为局部循环群的可解群的结构．     

On Uniformly Locally Compact Quasi-Uniform Hyperspaces

We characterize those Tychonoff quasi-uniform spaces for which the Hausdorff-Bourbaki quasi-uniformity is uniformly locally compact on the family of nonempty compact subsets of X. We deduce, among other results, that the Hausdorff-Bourbaki quasi-uniformity of the locally finite quasi-uniformity of a Tychonoff space Xis uniformly locally compact on if and only if Xis paracompact and locally compact. We also introduce the notion of a co-uniformly locally compact quasi-uniform space and show that a Hausdorff topological space is -compact if and only if its (lower) semi-continuous quasi-uniformity is co-uniformly locally compact. A characterization of those Hausdorff quasi-uniform spaces for which the Hausdorff-Bourbaki quasi-uniformity is co-uniformly locally compact on is obtained.     

Abstract Inverse Estimation with Application to Deconvolution on Locally Compact Abelian Groups

Recovery of the unknown parameter in an abstract inverse estimation model can be based on regularizing the inverse of the operator defining the model. Such regularized-inverse type estimators are constructed with the help of a version of the spectral theorem due to Halmos, after suitable preconditioning. A lower bound to the minimax risk is obtained exploiting the van Trees inequality. The proposed estimators are shown to be asymptotically optimal in the sense that their risk converges to zero, as the sample size tends to infinity, at the same rate as this lower bound. The general theory is applied to deconvolution on locally compact Abelian groups, including both indirect density and indirect regression function estimation.     

局部有限超拓扑的局部紧性

本文讨论了赋予局部有限拓扑的非空闲子集超空间的局部紧性.主要结果是:X正则,则其闭子集超空间局部紧当且仅当X可表示成一个紧空间与一个离散空间的拓扑和.     

Fell-拓扑的伪紧性（英文）

设X是拓扑空间，CL（X）表示X的所有非空闭子集的族，本文得到了下述结果：在CL（X）上的Fell-拓扑是伪肾的当且仅当X是feebly-紧或者非局部紧或者非σ－紧，由此得到了对于伪紧性不是闭遗传的两类新的拓扑空间。     

Fundamental Sets of Functions on Locally Compact Abelian Groups

Let G be a locally compact Abelian group, and let X be a compact set of G. Given a positive definite function ?: G × G → ? whose real part is continuous at neutral element of G, we research a necessary and sufficient setting for the linear span of the set {x ∈ X → ?(x ? y): y ∈ X} to be dense in C(X) in the topology of uniform convergence. The context treated that is abstract encompasses classical cases of the literature, while other examples are entirely new.     

Time Regularity for Random Walks on Locally Compact Groups

Let G be a compactly generated, locally compact group, and let T be the operator of convolution with a probability measure μ on G. Our main results give sufficient conditions on μ for the operator T to be analytic in L ^p (G), 1 < p < ∞, where analyticity means that one has an estimate of form for all n = 1, 2, ... in L ^p operator norm. Counterexamples show that analyticity may not hold if some of the conditions are not satisfied.     

Weak Mixing of Random Walks on Groups

Let G be a locally compact -compact group with right Haar measure m and a regular probability measure on G. We say that is weakly mixing if for all gL (G) and all fL ¹(G) with fdm=0 we have n ^{–1 n} _k=1| ^k *f,g|0. We show that is weakly mixing if and only if is ergodic and strictly aperiodic. To prove this we use and prove some results about unimodular eigenvalues for general Markov operators.     

Locally Compact Path Spaces

It is shown that the space X^[0,1], of continuous maps [0,1]X with the compact-open topology, is not locally compact for any space X having a nonconstant path of closed points. For a T₁-space X, it follows that X^[0,1] is locally compact if and only if X is locally compact and totally path-disconnected. Mathematics Subject Classifications (2000) 54C35, 54E45, 55P35, 18B30, 18D15.     

Weil Representations and Some Nonreductive Dual Pairs in Symplectic and Unitary Groups

We study the restriction of the Weil representations of symplectic and unitary groups to subgroups that are the centralizers of certain elements, and show that these are multiplicity free. This work is along the line of the Howe philosophy for so-called dual pairs in the symplectic and unitary groups, but our dual pairs are not reductive. We also obtain an explicit formula for the character of the restriction to the centralizer of a regular unipotent element.     

The Derived Picard Group is a Locally Algebraic Group

Let A be a finite-dimensional algebra over an algebraically closed field K. The derived Picard group DPic _K (A) is the group of two-sided tilting complexes over A modulo isomorphism. We prove that DPic _K (A) is a locally algebraic group, and its identity component is Out⁰ _K (A). If B is a derived Morita equivalent algebra then DPic _K (A)DPic _K (B) as locally algebraic groups. Our results extend, and are based on, work of Huisgen-Zimmermann, Saorín and Rouquier.     

Geometrizing Infinite Dimensional Locally Compact Groups

We study groups having invariant metrics of curvature bounded below in the sense of Alexandrov. Such groups are a generalization of Lie groups with invariant Riemannian metrics, but form a much larger class. We prove that every locally compact, arcwise connected, first countable group has such a metric. These groups may not be (even infinite dimensional) manifolds. We show a number of relationships between the algebraic and geometric structures of groups equipped with such metrics. Many results do not require local compactness.

     

Conjugately Dense Subgroups of Locally Finite Chevalley Groups of Lie Rank 1

Of interest are the subgroups of various groups which have nonempty intersection with each class of conjugate elements of the group under study. We call these subgroups conjugately dense and study Neumann's problem of describing them in the Chevalley groups over a field. The main theorem lists all conjugately dense subgroups of the Chevalley groups of Lie rank 1 over a locally finite field.     

局部强紧空间的Hoare空间与Smyth空间

本文主要讨论局部强紧空间的性质,特别是其Hoare空间和Smyth空间的性质,证明了T_0空间为局部强紧空间的当且仅当其Hoare空间为局部强紧空间,局部强紧空间的Smyth空间为C-空间.对于强局部紧空间,我们有类似的结论.