Ordered spaces, metric preimages, and function algebras

We consider the Complex Stone-Weierstrass Property (CSWP), which is the complex version of the Stone-Weierstrass Theorem. If X is a compact subspace of a product of three linearly ordered spaces, then X has the CSWP if and only if X has no subspace homeomorphic to the Cantor set. In addition, every finite power of the double arrow space has the CSWP. These results are proved using some results about those compact Hausdorff spaces which have scattered-to-one maps onto compact metric spaces.     

Complex function algebras and removable spaces

The compact Hausdorff space X has the Complex Stone-Weierstrass Property (CSWP) iff it satisfies the complex version of the Stone-Weierstrass Theorem. W. Rudin showed that all scattered spaces have the CSWP. We describe some techniques for proving that certain non-scattered spaces have the CSWP. In particular, if X is the product of a compact ordered space and a compact scattered space, then X has the CSWP if and only if X does not contain a copy of the Cantor set.     

Dense Near-Rings of Operators on Locally Convex Spaces

We consider the near-ring C(V) of all continuous operators on a locally convex space V. Like in the Theorem of Stone-Weierstrass the question arises which subnear-rings N have the property that every operator in C(V) can be approximated by elements of N on compact subsets of V. It is our aim to show that this can be achieved with certain primitive subnear-rings of C(V). For this we invoke a deep Theorem of Wielandt-Betsch on interpolation properties of primitive near-rings. We also stress the fact that such a Theorem of Stone-Weierstrass type can only be obtained in the context of near-rings.     

A methodology for the constructive approximation of nonlinear operators defined on noncompact sets

We consider the constructive approximation of a non-linear operator that is known on a bounded but not necessarily compact set. Our main result can be regarded as an extension of the classical Stone-Weierstrass Theorem and also shows that the approximation is stable to small

disturbances.

This problem arises in the modelling of real dynamical systems where an input-output mapping is known only on some bounded subset of the input space. In such cases it is desirable to construct a model of the real system with a complete input-output map that preserves, in some approximate sense, the known mapping. The model is normally constructed from an algebra of elementary continuous functions.

We will assume that the input space is a separable Hilbert space. To solve the problem we introduce a special weak topology and show that uniform continuity of the given operator in the weak topology provides an alternative compactness condition that is sufficient to justify the desired approximation.     

Stone-weierstrass theorems for group-valued functions

Constructive groups were introduced by Sternfeld in [6] as a class of metrizable groupsG for which a suitable version of the Stone-Weierstrass theorem on the group ofG-valued functionsC(X, G) remains valid. As a way of exploring the existence of such Stone-Weierstrass-type theorems in this context we address the question raised in [6] as to which groups are constructive and prove that a locally compact group with more than two elements is constructive if and only if it is either totally disconnected or homeomorphic to some vector group ℝ ⁿ . It may therefore be concluded that the Stone-Weierstrass theorem can be extended to some noncommutative Lie groups — exactly to those not containing any nontrivial compact subgroup. Research partially supported by Grant CTIDIB/2002/192 of theAgencia Valenciana de Ciencia y Tecnología, and Fundació Caixa-Castelló, grant P1 B2001-08.     

A Stone-Weierstrass theorem for Banach function spaces satisfying a certain separation property

We consider a strong lattice property for a Banach function space B on a compact Hausdorff space, which gives a general Stone-Weierstrass theorem for B. We also study the relation of this theorem and its proof to a certain decomposition of an associated compactification, and to another lattice-like property.     

Über den Typenkonvergenzsatz auf zusammenhängenden Lie Gruppen

The Convergence of Types Theorem on ^d is wellknown as an important tool for investigations on the limit behaviour of normalized sums or r.v. It is natural to look for a generalization for group-valued r.v. While for simply connected nilpotent Lie groups the Theorem is valid in general the existence of non-trivial compact subgroups causes problems. For compact extensions of nilpotent groups we prove restricted versions of the Convergence of Types Theorem.

---

---

Herrn Prof. Dr. L. Schmetterer zum 70. Geburtstag gewidmet     

F-空间中的一个基本定理及其在数值分析中的应用

本文针对F-空间中闭算子方程的一般逼近格式,研究其相容性、收敛性和稳定性之间的关系.所得的主要结果是：这种一般逼近格式在相容性条件下,其收敛性与稳定性是等价的.此定理可以看作是对Lax等价原理的推广,是求解第一类闭算子方程的一般逼近格式的基本定理.为得到这一主要结果,本文还给出了F-空间中的一条基本定理,众所周知的一致有界原理,闭图像定理和开映像定理是其简单推论.     

Maximum Matchings in the n-Dimensional Cube

The problem of efficient computation of maximum matchings in the n-dimensional cube, which is applied in coding theory, is solved. For an odd n, such a matching can be found by the method given in our Theorem 2. This method is based on the explicit construction (Theorem 1) of the maps of the vertex set that induce largest matchings in any bipartite subgraph of the n-dimensional cube for any n.     

Uniform factorization for compact sets of operators

We prove a factorization result for relatively compact subsets of compact operators using the Bartle and Graves Selection Theorem, a characterization of relatively compact subsets of tensor products due to Grothendieck, and results of Figiel and Johnson on factorization of compact operators. A further proof, essentially based on the Banach-Dieudonné Theorem, is included. Our methods enable us to give an easier proof of a result of W.H. Graves and W.M. Ruess.

     

Remainders in compactifications and generalized metrizability properties

When does a Tychonoff space X have a Hausdorff compactification with the remainder belonging to a given class of spaces? A classical theorem of Henriksen and Isbell and certain theorems, involving a new completeness type property introduced below, are applied to obtain new results on remainders of topological spaces and groups. In particular, some strong necessary conditions for a topological group to have a metrizable remainder, or a paracompact p-remainder, are established (the group itself turns out to be a paracompact p-space (Theorem 4.8)). It follows that if a non-locally compact topological group G is metrizable at infinity, then G is a Lindelöf p-space, and the Souslin number of G is countable (Corollary 4.10). This solves Problem 10.28 from [M. Hušek, J. van Mill (Eds.), Recent Progress in General Topology, vol. 2, North-Holland, 2002, pp. 1–57].     

Solvability of Rado systems in D-sets

Rado's Theorem characterizes the systems of homogeneous linear equations having the property that for any finite partition of the positive integers one cell contains a solution to these equations. Furstenberg and Weiss proved that solutions to those systems can in fact be found in every central set. (Since one cell of any finite partition is central, this generalizes Rado's Theorem.) We show that the same holds true for the larger class of D-sets. Moreover we will see that the conclusion of Furstenberg's Central Sets Theorem is true for all sets in this class.     

Equivariant Lefschetz maps for simplicial complexes and smooth manifolds

Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of Lefschetz numbers for self-maps to an equivariant K-homology class. We compute the Lefschetz invariants for self-maps of finite-dimensional simplicial complexes and smooth manifolds. The resulting invariants are independent of the extra structure used to compute them. Since smooth manifolds can be triangulated, we get two formulas for the same Lefschetz invariant in this case. The resulting identity is closely related to the equivariant Lefschetz Fixed Point Theorem of Lück and Rosenberg.     

Topological vector spaces, compacta, and unions of subspaces

In the first two sections, we study when a σ-compact space can be covered by a point-finite family of compacta. The main result in this direction concerns topological vector spaces. Theorem 2.4 implies that if such a space L admits a countable point-finite cover by compacta, then L has a countable network. It follows that if f is a continuous mapping of a σ-compact locally compact space X onto a topological vector space L, and fibers of f are compact, then L is a σ-compact space with a countable network (Theorem 2.10). Therefore, certain σ-compact topological vector spaces do not have a stronger σ-compact locally compact topology.In the last, third section, we establish a result going in the orthogonal direction: if a compact Hausdorff space X is the union of two subspaces which are homeomorphic to topological vector spaces, then X is metrizable (Corollary 3.2).     

Kaplansky Density and Kadison Transitivity Theorems for Irreducible Representations of Real C＊-Algebras

We prove analogs of the Kaplansky Density Theorem and the Kadison Transitivity Theorem for irreducible representations of a real C＊-algebra on a real Hilbert space. Specifically, if a C＊-algebra is acting irreducibly on a real Hilbert space, then the Hilbert space has either a real, complex, or quaternionic structure with respect to which the density and transitivity theorems hold.     

Ceva, Menelaus, and Selftransversality

The purpose of this paper is to state and prove a theorem (the CMS Theorem) which generalizes the familiar Ceva's Theorem and Menelaus' Theorem of elementary Euclidean geometry. The theorem concernsn -acrons (generalizations of n-gons) in affine space of any number of dimensions and makes assertions about circular products of ratios of lengths, areas, volumes, etc. In particular it contains, as special cases, many results in this area proved by earlier authors.     

Automorphism groups of compact projective planes

Compact connected projective planes have been investigated extensively in the last 30 years, mostly by studying their automorphism groups. It is our aim here to remove the connectedness assumption in some general results of Salzmann [31] and Hähl [14] on automorphism groups of compact projective planes. We show that the continuous collineations of every compact projective plane form a locally compact transformation group (Theorem 1), and that the continuous collineations fixing a quadrangle in a compact translation plane form a compact group (Corollary to Theorem 3). Furthermore we construct a metric for the topology of a quasifield belonging to a compact projective translation plane, using the modular function of its additive group (Theorem 2).     

Analytic density in Lie groups

A subgroupH of an analytic groupG is said to beanalytically dense if the only analytic subgroup ofG containingH isG itself. The main purpose of this paper is to give sufficient conditions onG (analogous to those of [8], [9], and [7] in the case of Zariski density) which guarantee the analytic density of cofinite volume subgroupsH. First we consider the case of arbitrary cofinite volume subgroups (Theorem 5 and its corollaries). Then we specialize to lattices, and prove the following result (Theorem 8):Let G be an analytic group whose radical is simply connected and whose Levi factor has no compact part and a finite center. Then any lattice in G is analytically dense. In proving this use is made of a result of Montgomery which also implies that for any simply connected solvable group, cocompactness of a closed subgroup implies analytic density. In the case of a solvable group with real roots this means analytic density and cocompactness are equivalent and thus completes a circle of ideas raised in Saito [13]. In Corollary 9 we deal with a related local condition. Finally in Theorem 10 and its corollaries we apply these considerations to prove a homomorphism extension theorem and an isomorphism theorem for 1-dimensional cohomology.     

A note on <Emphasis Type="Italic">tt</Emphasis><Superscript>*</Superscript>-bundles over compact nearly Kähler manifolds

First, we generalize a rigidity result for harmonic maps of Gordon (Gordon (1972) Proc AM Math Soc 33: 433–437) to generalized pluriharmonic maps. We give the construction of generalized pluriharmonic maps from metric tt ^*-bundles over nearly Kähler manifolds. An application of the last two results is that any metric tt ^*-bundle over a compact nearly Kähler manifold is trivial (Theorem A). This result we apply to special Kähler manifolds to show that any compact special Kähler manifold is trivial. This is Lu’s theorem (Lu (1999) Math Ann 313: 711–713) for the case of compact special Kähler manifolds. Further we introduce harmonic bundles over nearly Kähler manifolds and study the implications of Theorem A for tt ^*-bundles coming from harmonic bundles over nearly Kähler manifolds.     

Intégrabilité au sens de Cauchy et -intégrabilité pour des applications à valeurs dans un espace de Banach

The equivalence between the Cauchy left-integrability and the Riemann-integrability, for a bounded function defined on a compact interval of with values in a Banach space, is a particular case of Theorem 2.1. A first generalization to the case of functions defined on a compact rectangle of ² is given by Theorem 2.5.