Reflection principle and strong Markov property for multiparameter group-valued additive processes

On the discontinuity of the π1-action

Let (X,+) be a topological abelian group. We discuss regularity of solutions f:X→R of Hlawka’s functional inequality f(x+y)+f(y+z)+f(x+z)≤f(x+y+z)+f(x)+f(y)+f(z), postulated for all x,y,z∈X. We study the lower and upper hull of f. Moreover, we provide conditions which imply continuity of f. We prove, in particular, that if X is generated by any neighborhood of zero, f is continuous at zero, and f(0)=0, then f is continuous on X.

This article on Brownian motion starts with a short historical survey on its discovery and importance and continues with its most relevant properties. Mathematically detailed formulas are given and connections between the properties are indicated, however no derivations or proofs are provided. The properties covered start with the mathematical modeling of Brownian motion via the Wiener process; equivalent definitions and constructions such as the Lévy characterization, the midpoint displacement, and the rescaled random walks are discussed. Basic properties given include uniqueness, symmetry, restarted Brownian motion, and time inversion. More advanced topics treated cover stopping times, the strong Markov property, the reflection principle, extreme values, escape times from a strip, and last visits. Path oscillations, stationarity, Gaussian white noise, and Karhunen‐Löwe expansions are also discussed. Brownian motion with drift, the Brownian bridge, as well as intersections, recurrence, and transience of Brownian motion in higher dimensions, are mentioned briefly.

Notes on countable tightness of the subspaces of free (Abelian) topological groups

Let G be a reflexive topological group, and Gits group of characters, endowed with the compact open topology. We prove that the evaluation mapping from Gx G into the torus T is continuous if and only if G is locally compact. This is an analogue of a well-known theorem of Arens on admissible topologies on C(X) . DEFINITIONS AND REMARKS Let X, Y be topological spaces, and let Z be a subset of yX . A topology on Z is said to be admissible if the evaluation mapping from the product Z x X into Y, defined by w(f, x) = f(x), is continuous. Let (S, V) = {f E Z; f(S) C V}. The family {(S, V)}, where S runs over the collection of all compact subsets of X and V runs over a basis of open sets in Y, is a subbase for the compact open topology on Z. An admissible topology on Z must be finer than the compact open topology [8]. A result of Arens states that the existence of a coarsest admissible topology for the class of real continuous functions on a completely regular space X is equivalent to X being locally compact [1]. In this paper we are interested in reflexive topological groups. Answering in the negative a question of Megrelishvili [7], we will see that the evaluation map for those groups need not be continuous. In fact, as specified in the theorem, the continuity of the evaluation characterizes locally compact Hausdorff abelian groups among reflexive groups. We prove this result by means of convergence spaces. For an account of theory of convergence spaces the reader is referred to [3, 4]. We only give here needed definitions. Let G be a Hausdorff topological abelian group, and let FG be the set of all continuous homomorphisms from G into the unidimensional torus T. If addition is defined pointwise in FG, it becomes an abelian group. The continuous Received by the editors October 5, 1993 and, in revised form, April 25, 1994; the contents of this paper were presented to the First Ibero-American Conference on Topology and its Applications in Benicassim, Spain, on March 28-30, 1995. 1991 Mathematics Subject Classification. Primary 22A05.

Long colimits of topological groups II: Free groups and vector spaces

We introduce, for any topological abelian group G, the property of equicontinuity of arcs of \(G^\wedge _p\), the dual group of G endowed with its pointwise topology. We analyze the implications of this property, which we denote by EAP\(_\sigma \), and we present some representative examples. Furthermore we prove that if G satisfies EAP\(_\sigma \), every element of the arcwise connected component of \(G^\wedge _p\) can be written as \(\phi (1)\) for a suitable one-parameter subgroup \(\phi :\mathbb {R} \rightarrow G^\wedge _p\).

In reminiscence of Ptak's open mapping theorem, a topological space satisfying the open mapping theorem is called a B r space.This paper is devoted to the study of sums and products of B r spaces in the category of topological spaces.We prove that, in general, sums and products of even two B r spaces need no longer be B r .On the other hand, for any B r space E the sum E θ E is again a B r space.Moreover, since Cech complete spaces are known to be B r , we ask whether a sum E θ F is B r provided that E is Cech complete and F is B r .It turns out that, at least in the framework of complete regularity, the answer to this question is in the positive if and only if F is a Baire space.Introduction.The notion of B r spaces has first been introduced by T. Husain in the categories of locally convex vector spaces (see [HuJ) and topological groups (see [Hu 2 ]).Originally, it goes back to V. Ptak's open mapping theorem (see [Kό], p. 35 ff).A Hausdorff topological space E is called a B r space if every continuous, nearly open bijection / from E onto any Hausdorff space F is in fact open.For a survey of the classical theory of B r spaces we refer the reader to Kόthe's book [Kό], where the locally convex case is treated.The linear topological case is investigated in the lecture notes [AEK].B r groups are considered by several authors.See for instance [Hu 2 ], [Ba], [Gr], [Su].In a purely topological context, B r spaces have first been investigated by Weston in [We], although the term B r space is not used there.Translated into the B r terminology, Weston proved that every completely metrizable topological space is a B r space.His result has been generalized by Byczkowski and Pol in [BP], who proved that every Cech complete topological space is a B r space.In [No] we have further generalized this result proving that every Hausdorff, semi-regular topological space densely containing some Cech complete sub space is in fact a B r space.We just mention another generalization of Byczkowski and Pols' result into a somewhat different direction by Wilhelm (cf.[Wi]).In the present paper we examine the invariance of the class of B r spaces under topological sums and products.It turns out that, in general, the sum of even two B r spaces need not be a B r space.A counterexample is given in §2.In §1 we obtain a positive result stating that the sum E θ E

Chapter 23 - Normed Operators

I. V. M. Klimkin, "On extension of a vector measure," Izv. Vyssh. Uchebn. Zaved., Mat., ~, 46-53 (1971). 2. V. M. Klimkin, "On certain properties of continuous outer measures," Uch. Zap. Leningr. Gos. Pedagog. Inst., 469, 187-204 (1970). 3. N. S. Gusel'nikov, "On the convergence of a sequence of continuous triangular measures," Nauch. Dokl., Mat., 26, 78-82, Hercen Lectures, Izd. Mosk. 0blast. Pedagog. Inst. im N. K. Krypskoi, Moscow (1973). 4. N. S. Gusel'nikov, "On the extension of quasi-Lipschitz set functions," Mat. Zametki, 17, No. i, 21-31 (1975). 5. N. Dunford and J. T. Schwartz, Linear Operators, Part I, Wiley, New York (1958). 6. J. L. Kelley, General Topology, Van Nostrand, New York (2955). 7. L. Ya. Savel'ev, "On the extension of a mappin E of a Boolean ring into a topological Abelian group," Dokl. Akad. Nauk SSSR, 149, No. 6, 1268-1269 (1963). 8. L. Ya. Savel'ev, "Extension of measures by continuity," Sib. Mat. Zh., ~, No. 3, 639650 (1964). 9. L. Ya. Savel'ev, "Certain conditions for extendability of vector measures," Sib. Mat. Zh., 9, No. 4, 937-950 (1968). 10. M. Kh. Khafizov, "On the extension of a vector-valued measure," in: Current Problems of Mathematical Logic and Set Theory [in Russian], Izd. Mosk. Gos. Pedagog. Inst., Moscow (1975), pp. 291-301. ii. P. R. Halmos, Measure Theory, Van Nostrand, New York (1950).

A class \( \mathcal{G} \) of maximally almost periodic (MAP) topological Abelian groups is called a UMAP-class if it has the following property: if G is an Abelian group and τ 1 and τ 2 are distinct group topologies in G such that (G, τ 1) and (G, τ 2) are in \( \mathcal{G} \), then (G, τ 1)∧ ≠ (G, τ 2)∧. Several examples of UMAP-classes are discussed. In particular, it is shown that the class PMAP of all Polish MAP-groups is a UMAP-class.

Bredon homology and ramified covering G-maps

We deal with the Krein-Langer problem for\(\mathcal{L}(\mathcal{H})\)-valued functions on the band (−2a, 2a)×Γ, where\(\mathcal{L}(\mathcal{H})\) is the algebra of continuous linear operators on a Hilbert space\(\mathcal{H}\),a a finite positive number and Γ a topological Abelian group. We show that every weakly continuous κ-indefinite function\(f:( - 2a,2a) \times \Gamma \to \mathcal{L}(\mathcal{H})\) admits a strongly continuous κ-indefinite continuation to ℝ × Γ with the same indefiniteness index κ. We give a parametrization of the extensions in terms of operator-valued Schur functions.

A note on multiplier convergent series

Introduction. This article is based upon a principle which is so standard that it is almost a cliche: The first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation. To be more specific, we shall describe some applications of this to the symmetry questions of topological transformation groups. Given a topological space X, let Homeo(^) denote the set of self-homeomorphisms of X. This is a group under composition of mappings. If G is an arbitrary group, then a group action of G on I is a homomorphism Homeo( X). Frequently we wish to impose some weak assumptions on <p. For example, if G is a topological group, then we might want cp to have suitable continuity properties. The usual assumption is that the map