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This paper is produced in response to the questioning of Morse decomposition for semigroup actions on noncompact spaces. We show how the limit behavior can be studied in arbitrary topological spaces by using powerful tools such as the Stone-?Cech compactification and shadowing semigroups. We extend Conley's characterization of chain recurrence in terms of attractors from the setting of flows on compact metric spaces to the setting of semigroup actions on any topological space. [ABSTRACT FROM AUTHOR]

We extend the definition of Bockstein basis $sigma (G)$ to nilpotent groups $G$. A metrizable space $X$ is called a {it Bockstein space} if $operatorname {dim}_G(X) = sup {operatorname {dim}_H(X) | Hin sigma (G)}$ for all Abelian groups $G$. The Bockstein First Theorem says that all compact spaces are Bockstein spaces. par Here are the main results of the paper: vspace {4pt} par noindent {bf Theorem 0.1.} {em Let $X$ be a Bockstein space. If $G$ is nilpotent, then $operatorname {dim}_G(X) leq 1$ if and only if $sup {operatorname {dim}_H(X) | Hin sigma (G)}leq 1$.} vspace {3pt} par noindent {bf Theorem 0.2.} {em $X$ is a Bockstein space if and only if $operatorname {dim}_{{mathbf {Z}}_{(l)}} (X) = operatorname {dim}_{hat {Z}_{(l)}}(X)$ for all subsets $l$ of prime numbers.} [ABSTRACT FROM AUTHOR]

Let $S$ be a compact space and let $X$, $\left| \cdot \right| _{X}$ be a (real, for simplicity) Banach space. We consider the space $C_{X}=C\left( S,X\right) $ of all continuous $X$-valued functions on $S$, with the supremum norm $\left\| \cdot \right| _{\infty }$. We prove in this paper a Bochner integral representation theorem for bound\-ed linear operators \begin{equation*} T:C_{X}\longrightarrow X \end{equation*} which satisfy the following condition: \begin{equation*} x^{*},y^{*}\in X^{*},f,g\in C_{X}:x^{*}\circ f=y^{*}\circ g\Longrightarrow x^{*}\circ Tf=y^{*}\circ Tg \end{equation*} where $X^{*}$ is the conjugate space of $X$. In the particular case where $X=\mathbb{R}$, this condition is obviously satisfied by every bounded linear operator \begin{equation*} T:C_{\mathbb{R}}\longrightarrow \mathbb{R} \end{equation*} and the result reduces to the classical Riesz representation theorem. If the dimension of $X$ is greater than $2$, we show by a simple example that not every bounded linear $T:C_{X}\longrightarrow X$ admits an integral representation of the type above, proving that the situation is different from the one dimensional case. Finally we compare our result to another representation theorem where the integration process is performed with respect to an operator valued measure. [ABSTRACT FROM AUTHOR]

In this paper the $Lip\alpha$ harmonic approximation ($0 < \alpha < \frac {1}{2}$) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on $Lip\alpha$ harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in $Lip\alpha $ harmonic approximation are also studied. [ABSTRACT FROM AUTHOR]

In this paper we determine the structure of the minimal ideal in the enveloping semigroup for the natural action of a connected semisimple Lie group on its maximal compact subgroup. In particular, if $G= KAN$ is an Iwasawa decomposition of the group $G$, then the group in the minimal left ideal is isomorphic both algebraically and topologically with the normalizer $M$ of $AN$ in $K.$ Complete descriptions are given for the enveloping semigroups in the cases $G=\text{\rm {SL}}(2, \mathbb{C})$ and $G=\text{\rm {SL}}(2, \mathbb{R}).$ [ABSTRACT FROM AUTHOR]

In this paper (weakly) separating maps between spaces of bounded continuous functions over a nonarchimedean field $\mathbb{K}$ are studied. It is proven that the behaviour of these maps when $\mathbb{K}$ is not locally compact is very different from the case of real- or complex-valued functions: in general, for $\mathbb{N}$-compact spaces $X$ and $Y$, the existence of a (weakly) separating additive map $T: \cx \ra \cy$ implies that $X$ and $Y$ are homeomorphic, whereas when dealing with real-valued functions, this result is in general false, and we can just deduce the existence of a homeomorphism between the Stone-Cech compactifications of $X$ and $Y$. Finally, we also describe the general form of bijective weakly separating linear maps and deduce some automatic continuity results. [ABSTRACT FROM AUTHOR]

This paper concerns an index theory for $\Bbb Z$-actions induced by a homeomorphism of a compact space. We give a definition of a genus for uniform spaces and prove that the genus for compact spaces is an index. To this end we show a ${\Bbb Z}$-version of the Borsuk-Ulam theorem and the existence of a continuous equivariant extension for these $\Bbb Z$-actions. [ABSTRACT FROM AUTHOR]