On <Emphasis Type="Italic">k</Emphasis>-Spaces and <Emphasis Type="Italic">k</Emphasis><Subscript><Emphasis Type="Italic">R</Emphasis></Subscript>-Spaces

In this note we study the relation between k _R -spaces and k-spaces and prove that a k _R -space with a σ-hereditarily closure-preserving k-network consisting of compact subsets is a k-space, and that a k _R -space with a point-countable k-network consisting of compact subsets need not be a k-space. This work was supported by the NSF of China (10271056).     

Prime ideals in the C*-algebra of a nilpotent group

We say that a locally compact groupG hasT ₁ primitive ideal space if the groupC ^*-algebra,C ^*(G), has the property that every primitive ideal (i.e. kernel of an irreducible representation) is closed in the hull-kernel topology on the space of primitive ideals ofC ^*(G), denoted by PrimG. This means of course that every primitive ideal inC ^*(G) is maximal. Long agoDixmier proved that every connected nilpotent Lie group hasT ₁ primitive ideal space. More recentlyPoguntke showed that discrete nilpotent groups haveT ₁ primitive ideal space and a few month agoCarey andMoran proved the same property for second countable locally compact groups having a compactly generated open normal subgroup. In this note we combine the methods used in [3] with some ideas in [9] and show that for nilpotent locally compact groupsG, having a compactly generated open normal subgroup, closed prime ideals inC ^*(G) are always maximal which implies of course that PrimG isT ₁.     

Positive linear extension operators for spaces of affine functions

The following result is proved: letE be anF-space (that is, the space of all continuous affine functions defined on a compact universal cap van shing at zero) and letMχE be anM-ideal. Then, ifE/M is a π₁ with positive defining projections, then there is a positive linear operator ϱ:E/M→E of norm one such that ϱ lifts the canonical mapE→E/M. In the proof, which heavily depends on work of Ando, we study ensor products of certain convex cones with compact bases, and we calculate the norm of a positive linear operator defined on a finite dimensional space with range in aF-space. Various corollaries are deduced for split faces of compact convex sets and for morphisms ofC ^*-algebras.     

L 2-lower bounds for a special class of random walks

Summary We investigate theL ₂-speed of convergence to stationarity for a certain class of random walks on a compact connected Lie group. We give a lower bound on the number of stepsk necessary such that thek-fold convolution power of the original step distribution has anL ₂-density. Our method uses work by Heckman on the asymptotics of multiplicities along a ray of representations. Several examples are presented.This paper is based on parts of the author's doctoral dissertation written at The Johns Hopkins University     

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.     

On characterizingL 1-supports,natural and Chebyshev approximates

Characterizations of optimal linear supports and approximates in realn-dimensional space are derived. We first characterize anL ₁-support; that is, a support hyperplane for a continuous function, over a solid, compact set, which is optimal in theL ₁-sense. We then characterize best approximates; that is, affinely linear functions which are the best approximates to a continuous function on a compact set by some optimality criterion. The characterization is first derived under the Natural criterion and then expanded to the Chebyshev criterion, utilizing Geoffrion's Criterion Equivalence.     

On the theory of groups with generalized minimality condition for closed subgroups

We prove that a topological Abelian locally compact group with generalized minimality condition for closed subgroups is a group of one of the following types: 1) a group with minimality condition for closed subgroups, 2) an additive group of theJ _p -ring of integerp-adic numbers, 3) an additive groupR _p of the field ofp-adic numbers (p is a prime number). Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 398–409, March, 1999.     

Extension of locally pseudocompact group actions

It is shown that: (1) any action of a Moscow group G on a first countable, Dieudonné complete (in particular, on a metrizable) space X can uniquely be extended to an action of the Dieudonné completion γG on X, (2) any action of a locally pseudocompact topological group G on a b _f -space (in particular, on a first countable space) X can uniquely be extended to an action of the Weil completion on the Dieudonné completion γX of X. As a consequence, we obtain that, for each locally pseudocompact topological group G, every G-space with the b _f -property admits an equivariant embedding into a compact Hausdorff G-space. Furthermore, for each pseudocompact group G, every metrizable G-space has a G-invariant metric compatible with its topology. We also give a direct construction of such an invariant metric. Received: June 22, 2000; in final form: May 22, 2001?Published online: June 11, 2002     

Examples of sets with given approximation properties in WCG-space

An example of a nontrivial convex bounded Chebyshev set is constructed in an arbitrary weakly compactly generated Banach space (WCG-space). An example of bounded approximately compact but not locally compact set is constructed in an arbitrary infinite-dimensional WCG-space.     

A note on representation of pseudovariant maps

We define and study the notions of pseudovariant maps and pseudogeny maps on G-spaces. We prove that the set of all pseudogenies of a locally comapct G-space is a group. We further obtain representation of pseudovariant maps in terms of pseudogenies. Finally, we obtain Tietze type extension result for pseudovariant homotopies defined on locally compact second countable G-spaces.     

Topological Rings of Quotients and Rings Without Open Left Ideals

Simple locally compact rings without open left ideals were considered in [13] and general locally compact rings without open left ideals were studied extensively in [5] and [6]. We remove the hypothesis of local compactness and consider topological rings A without open left ideals but containing an open subring R. In section 4 we show that under these conditions A is completely determined by R. More precisely A can be identified with the topological ring of quotients C(R) introduced in [8]. As an R-module _RA is topologically isomorphic to I_*(_RR), the topological injective hull of _RR. The last statement was proved in [6] for A locally compact and R compact. Section 5 gives a characterization of those linearly topologized rings R that can be openly embedded into a ring A without open left ideals. In particular we shall show that the open left ideals form an idempotent ideal filter with quotient ring A. In section 6 we consider the class ? of all topological rings that can be openly embedded into a topological ring without open left ideals. If we restrict our attention to linearly topologized rings, then ? is Morita-invariant. In section 2 we construct a topological ring of quotients Q_*(R) and prove that it coincides with the ring C(R) of [8].     

On embedding of locally compact abelian topological groups in euclidean spaces. I

All the groups considered in this paper are locally compact abelianT ₀-topological groups with countable bases. They are referred to as LCA-groups. For each positive integerm, R ^m denotes them-dimensional euclidean space.Research supported by Hungarian National Foundation for Scientific Research Grant No. 27-3-232.     

The space of maps from a locally compact space to a Banach space

The space of continuous maps from a topological spaceX to topological spaceY is denoted byC(X,Y) with the compact-open topology. In this paper we prove thatC(X,Y) is an absolute retract ifX is a locally compact separable metric space andY a convex set in a Banach space. From the above fact we know thatC(X,Y) is homomorphic to Hilbert spacel ₂ ifX is a locally compact separable metric space andY a separable Banach space; in particular,C(R ⁿ,R^m) is homomorphic to Hilbert spacel ₂. This research is supported by the Science Foundation of Shanxi Province's Scientific Committee     

Affine complete locally convex hypersurfaces

An open problem in affine geometry is whether an affine complete locally uniformly convex hypersurface in Euclidean (n+1)-space is Euclidean complete for n≥2. In this paper we give the affirmative answer. As an application, it follows that an affine complete, affine maximal surface in R ³ must be an elliptic paraboloid. Oblatum 16-VI-2001 & 27-II-2002?Published online: 29 April 2002     

Faserungen,die Aus Reguli mit Einem Gemeinsamen Geradenpaar Zusammengesetzt Sind

Let _pp be a pappian projective 3-space and be a set of lines of _pp; we define:-a lineg of _pp has theproperty R with respect to C, if all lines of meetingg form a regulus- has theproperty E ₂, if there exists a pencil of lines containing only lines having the propertyR with respect to . A spread having the propertyE ₂ is built up of reguli which have two lines in common. In the present paper, we point out a method of constructing a spread of _pp which has the propertyE ₂. This construction is applied to the real 3-space to produce an example of a spread which has the property E₂ and is different from an elliptic linear line congruence; the discussed example represents a 4-dimensional translation plane with a 6-dimensional collineation group. Finally, the propertiesE ₂ andR are used to characterize the elliptic linear line congruences of a pappian 3-space.     

Orthogonal hyperspheres

Umbilical projection ([12], [14]) is a process suggested to derive results rather quickly in regard to four intersecting spheres [17] andn+1 intersecting hyperspheres in ann-space [18]. The same has been used with an advantage to deduce a porism on 2n+5 hyperspheres in ann-space [23]. The purpose of this paper is to concentrate on mutually orthogonal hyperspheres only and to illustrate simultaneously once again the utility and facility of this tool to arrive at a number of new and interesting results as follows:The 2(n+1) intersections ofn+1 mutually orthogonal hyperspheres in ann-space, takenn at a time, give rise to 2 ⁿ pairs ofsemi-inverse [22] simplexes, perspective from their radical centreH, such that the 2 ⁿ primes of perspectivity coincide with their 2 ⁿ hyperplanes of similitude and form anS-configuration (S-C) [15] with theircentral simplex S(A) as itsdiagonal simplex. Everysimplex of intersection introduced here isisodynamic [25] such that itstangential simplex, circumscribed to it along circumhypersphere, is perspective to it from itsLemoine point L. ItsLemoine hyperplane l, as the polar prime ofL w. r. t. it, is the same as that of itscomplementary simplex of intersection and coincides whith their prime of perspectivity such that their 2(n+1) altitudes are met by their commonBrocard diameter through their Lemoine points. The 2 ⁿ Brocard diameters of the 2 ⁿ pairs of complementary simplexes of intersection concur atH. The hyperspheres of antisimilitude of the given hyperspheres, having centres in a prime of similitude, form the commonNeuberg hyperspheres of the pair of semi-inverse simplexes, having this prime as their common Lemoine hyperplane, are consequently orthogonal to their cirumhyperspheres whose radical hyperplane, too, coincides whith this prime, and therefore belong to acoaxal net [15] passing through the pair of their commonNeuberg points on their common Brocard diameter. The second centres of similitude of the 2 ⁿ pairs ofcomplementary hyperspheres of intersection form the 2 ⁿ vertices of the dual [15] of the (S-C), whithS(A) as common diagonal simplex, as its polar reciprocal w. r. t. the common orthogonal hypersphere of then+1 hyperspheres, the first centres of similitude coinciding atH.Due inspiration is derived from the works ofCourt ([2]–[9]) on mutually orthogonal circles and spheres. Presented by G. Hajós     

On locally symmetric vector fields on Riemannian manifolds

It is shown that if ann-dimensional (n≧3) Riemannian manifold admitsr≧2 locally symmetric vector fields (LSVF's), then it is aV(k)-space. In particular, ifr=n−1 then the manifold is a space of constant curvature. In the case of a 3-dimensional Riemannian manifold a close connection between LSVF's and eigenvectors of the Ricci tensor is found.     

The wΔ-space problem

We address the following question: “Must every wΔ-space with a G_δ-diagonal be developable?” Consistently, the answer is “no.”Example. Assume CH. There is a zero-dimensional, scattered, locally compact, wΔ-space with a G_δ-diagonal which is not developable.For normal, locally compact spaces (or slightly weaker), the answer is “yes”.Theorem. If X is ω-sCWH, locally Lindelöf, wΔ-space with a G_δ-diagonal, then X is developable.     

Unified characterizations for modifications ofR 0 andR 1 topological spaces

Cammaroto and Noiri [14] introduced a separation axiom calledm-R ₀ in anm-space (X, m). In this paper, we introduce the notion ofm-R ₁ spaces and offer many characterizations ofm-R ₀ (resp.m-R ₁) spaces which enable us to obtain unified characterizations of separation axiomsR ₀, semi-R ₀, pre-R ₀,α-R ₀,δ-semiR ₀, (δ, p)-R ₀ (resp.R ₁, semi-R ₁, pre-R ₁,α-R ₁,δ-semiR ₁, (δ, p)-R ₁).     

Invariant pseudometrics on Palais proper G-spaces

Let G be a locally compact Hausdorff group. It is proved that: 1. on each Palais proper G-space X there exists a compatible family of G-invariant pseudometrics; 2.the existence of a compatible G-invariant metric on a metrizable proper G-space X is equivalent to the paracompactness of the orbit space X/G; 3. if in addition G is either almost connected or separable, and X is locally separable, then there exists a compatible G-invariant metric on X. This revised version was published online in June 2006 with corrections to the Cover Date.