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## Über dieses Buch

This book is the result of a meeting on Topology and Functional Analysis, and is dedicated to Professor Manuel López-Pellicer's mathematical research. Covering topics in descriptive topology and functional analysis, including topological groups and Banach space theory, fuzzy topology, differentiability and renorming, tensor products of Banach spaces and aspects of Cp-theory, this volume is particularly useful to young researchers wanting to learn about the latest developments in these areas.

## Inhaltsverzeichnis

### On the Mathematical Work of Professor Manuel López-Pellicer

In Honour of Manuel López-Pellicer
Abstract
We examine selected topics of the research work of professor Manuel López-Pellicer. After an introductory section, the paper is divided in four main sections, which include his publications on Set Topology, Locally Convex Space Theory, $$C_{p}$$-theory and Descriptive Topology. We shall also glance at his work on Popular Mathematics.
Juan Carlos Ferrando

### A Note on Nonautonomous Discrete Dynamical Systems

In Honour of Manuel López-Pellicer
Abstract
A discrete nonautonomous dynamical system (a NDS for short) is a pair $$(X,f_{1,\infty })$$ where X is a topological space and $$f_{1,\infty }$$ is a sequence of continuous functions $$\left( f_{1},f_{2},\ldots \right)$$ from X to itself. The orbit of a point $$x \in X$$ is defined as the set $$\mathscr {O}_{f_{1,\infty }}(x) := \{x, f_{1}^{1}(x), f_{1}^{2}(x), \ldots , f_{1}^{n}(x), \ldots \}$$. NDS’ were introduced by S. Kolyada and L. Snoha in 1996 and they are related to several mathematical fields; among others the theory of difference equations. Notice that NDS’ generalize the usual notion of a discrete dynamical system: indeed, if suffices to take $$f_{1,\infty }$$ as a constant sequence. The aim of this note is twofold. First we analyze several definitions of a periodic point in the framework of NDS’. The interest of this notion comes from the fact that, in the realm of discrete dynamical systems, the third condition of the definition of Devaney’s chaos (sensitivity) follows from the first two (transitivity and the set of periodic points is dense). This result need not be true for NDS’ and the results in this context depend upon the definition of a periodic point we consider. Secondly, we present several results on transitivity. In contrast to the situation for discrete dynamical systems, there exists second countable, perfect metric NDS’ with the Baire property which have transitive points but they are not transitive. Among other things, we study the relationships between these two notions.
Gerardo Acosta, Manuel Sanchis

### Linear Operators on the (LB)-Sequence Spaces

In Honour of Manuel López-Pellicer
Abstract
We determine various properties of the regular (LB)-spaces $$ces(p-)$$, $$1<p\le \infty$$, generated by the family of Banach sequence spaces $$\{ces(q):1<q<p\}$$. For instance, $$ces(p-)$$ is a (DFS)-space which coincides with a countable inductive limit of weighted $$\ell _1$$-spaces; it is also Montel but not nuclear. Moreover, $$ces(p-)$$ and $$ces(q-)$$ are isomorphic as locally convex Hausdorff spaces for all choices of $$p, q\in (1,\infty ]$$. In addition, with respect to the coordinatewise order, $$ces(p-)$$ is also a Dedekind complete, reflexive, locally solid, lc-Riesz space with a Lebesgue topology. A detailed study is also made of various aspects (e.g., the spectrum, continuity, compactness, mean ergodicity, supercyclicity) of the Cesàro operator, multiplication operators and inclusion operators acting on such spaces (and between the spaces $$\ell _{r-}$$ and $$ces(p-)$$).
Angela A. Albanese, José Bonet, Werner J. Ricker

### Equicontinuity of Arcs in the Pointwise Dual of a Topological Abelian Group

In Honour of Manuel López-Pellicer
Abstract
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 $$\phi :\mathbb {R} \rightarrow G^\wedge _p$$.
María Jesús Chasco, Xabier Domínguez

### On Ultrabarrelled Spaces, their Group Analogs and Baire Spaces

In Honour of Manuel López-Pellicer, Loyal Friend and Indefatigable Mathematician
Abstract
Let E and F be topological vector spaces and let G and Y be topological abelian groups. We say that E is sequentially barrelled with respect to F if every sequence $$(u_n)_{n\in \mathbb {N}}$$ of continuous linear maps from E to F which converges pointwise to zero is equicontinuous. We say that G is barrelled with respect to F if every set $$\mathscr {H}$$ of continuous homomorphisms from G to F, for which the set $$\mathscr {H}(x)$$ is bounded in F for every $$x\in E$$, is equicontinuous. Finally, we say that G is g-barrelled with respect to Y if every $$\mathscr {H}\subseteq \mathrm{CHom}(G,Y)$$ which is compact in the product topology of $$Y^ G$$ is equicontinuous. We prove that
• a barrelled normed space may not be sequentially barrelled with respect to a complete metrizable locally bounded topological vector space,
• a topological group which is a Baire space is barrelled with respect to any topological vector space,
• a topological group which is a Namioka space is g-barrelled with respect to any metrizable topological group,
• a protodiscrete topological abelian group which is a Baire space may not be g-barrelled (with respect to $$\mathbb R/\mathbb Z$$).
We also formulate some open questions.

### Subdirect Products of Finite Abelian Groups

In Honour of Manuel López-Pellicer
Abstract
A subgroup G of a product $$\prod \limits _{i\in \mathbb {N}}G_i$$ is rectangular if there are subgroups $$H_i$$ of $$G_i$$ such that $$G=\prod \limits _{i\in \mathbb {N}}H_i$$. We say that G is weakly rectangular if there are finite subsets $$F_i\subseteq \mathbb {N}$$ and subgroups $$H_i$$ of $$\bigoplus \limits _{j\in F_i} G_j$$ that satisfy $$G=\prod \limits _{i\in \mathbb {N}}H_i$$. In this paper we discuss when a closed subgroup of a product is weakly rectangular. Some possible applications to the theory of group codes are also highlighted.

### Maximally Almost Periodic Groups and Respecting Properties

In Honour of Manuel López-Pellicer
Abstract
A maximally almost periodic topological (MAP) group G respects $$\mathscr {P}$$ if $$\mathscr {P}(G)=\mathscr {P}(G^+)$$, where $$G^+$$ is the group G endowed with the Bohr topology and $$\mathscr {P}$$ stands for the subsets of G that have the property $$\mathscr {P}$$. For a Tychonoff space X, we denote by $$\mathfrak {P}$$ the family of topological properties $$\mathscr {P}$$ of being a convergent sequence or a compact, sequentially compact, countably compact, pseudocompact and functionally bounded subset of X, respectively. We study relations between different respecting properties from $$\mathfrak {P}$$ and show that the respecting convergent sequences ($$=$$the Schur property) is the weakest one among the properties of $$\mathfrak {P}$$. We characterize respecting properties from $$\mathfrak {P}$$ in wide classes of MAP topological groups including the class of metrizable MAP abelian groups. Every real locally convex space (lcs) is a quotient space of an lcs with the Schur property, and every locally quasi-convex (lqc) abelian group is a quotient group of an lqc abelian group with the Schur property. It is shown that a reflexive group G has the Schur property or respects compactness iff its dual group $$G^\wedge$$ is $$c_0$$-barrelled or g-barrelled, respectively. We prove that an lqc abelian $$k_\omega$$-group respects all properties $$\mathscr {P}\in \mathfrak {P}$$. As an application of the obtained results we show that a reflexive abelian group of finite exponent is a Mackey group.
Saak Gabriyelyan

### Forty Years of Fuzzy Metrics

In Honour of Manuel López-Pellicer
Abstract
Kramosil and Michalek gave in 1975 a concept of fuzzy metric M on a set X which extends to the fuzzy setting the concept of probabilistic metric space introduced by K. Menger. After, George and Veeramani (Fuzzy Sets Syst 64: 395–399, 1994) modified the previous concept and gave a new definition of fuzzy metric. In both cases the fuzzy metric M induces a topology $$\tau _M$$ on X which is metrizable. In this paper we survey some results relative to both concepts. In particular, we focus our attention in the completion of fuzzy metrics in the sense of George and Veeramani, since there is a significative difference with respect to the classical metric theory (in fact, there are fuzzy metric spaces, in this sense, which are not completable), and also in fixed point theory in both senses because it is a high activity area.
Valentín Gregori, Almanzor Sapena

### Differentiability and Norming Subspaces

In Honour of Manuel López-Pellicer
Abstract
This is a survey around a property (Property $$\mathscr {P}$$) introduced by M. Fabian, V. Zizler, and the third named author, in terms of differentiability of the norm. Precisely, a Banach space X is said to have property $${\mathscr {P}}$$ if for every norming subspace $$N\subset X^*$$ there exists an equivalent Gâteaux differentiable norm for which N is 1-norming. Every weakly compactly generated space has property $${\mathscr {P}}$$. Applications to measure theory, the classification of compacta, and some other structural properties of compact and Banach spaces are given. Some open problems are listed, too. It is based on an earlier paper by Fabian, Zizler, and the third named author, and a recent one by the authors of the survey.
Antonio José Guirao, Aleksei Lissitsin, Vicente Montesinos

### Separable (and Metrizable) Infinite Dimensional Quotients of and Spaces

In Honour of Manuel López-Pellicer
Abstract
The famous Rosenthal-Lacey theorem states that for each infinite compact set K the Banach space C(K) of continuous real-valued functions on a compact space K admits a quotient which is either an isomorphic copy of c or $$\ell _{2}$$. Whether C(K) admits an infinite dimensional separable (or even metrizable) Hausdorff quotient when the uniform topology of C(K) is replaced by the pointwise topology remains as an open question. The present survey paper gathers several results concerning this question for the space $$C_{p}(K)$$ of continuous real-valued functions endowed with the pointwise topology. Among others, that $$C_{p}(K)$$ has an infinite dimensional separable quotient for any compact space K containing a copy of $$\beta \mathbb {N}$$. Consequently, this result reduces the above question to the case when K is a Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of $$\beta \mathbb {N}$$). On the other hand, although it is unknown if Efimov spaces exist in ZFC, we note under $$\lozenge$$ (applying some result due to R. de la Vega), that for some Efimov space K the space $$C_{p}(K)$$ has an infinite dimensional (even metrizable) separable quotient. The last part discusses the so-called Josefson–Nissenzweig property for spaces $$C_{p}(K)$$, introduced recently in [3], and its relation with the separable quotient problem for spaces $$C_{p}(K)$$.
Jerzy Kąkol

### Multiple Tensor Norms of Michor’s Type and Associated Operator Ideals

In Honour of Manuel López-Pellicer
Abstract
We study the $$(n+1)$$-tensor norms of Michor type and characterize the n-linear mappings of the components of its associated operator ideals.
Juan Antonio López Molina

### On Compacta K for Which C(K) Has Some Good Renorming Properties

In Honour of Manuel López-Pellicer
Abstract
By renorming it is usually meant obtaining equivalent norms in a Banach space with better properties, like being local uniformly rotund (LUR) or Kadets. In these notes we are concerned with C(K) spaces and pointwise lower semicontinuous Kadets or LUR renormings on them. If a C(K) space admits some of such equivalent norms then this space, endowed with the pointwise topology, has a countable cover by sets of small local norm-diameter (SLD). This property may be considered as the topological baseline for the existence of a pointwise lower semicontinuous Kadets, or even LUR renorming, since in many concrete cases it is the first step to obtain such a norm. In these notes we survey some methods, appearing in the literature, to prove that some C(K) spaces have this property.
Aníbal Moltó

### A Fixed Point Theory Linked to the Zeros of the Partial Sums of the Riemann Zeta Function

In Honour of Manuel López-Pellicer
Abstract
For each $$n>2$$ we consider the corresponding $$n\mathrm{th}$$-partial sum of the Riemann zeta function $$\zeta _{n}(z):=\sum _{j=1}^{n}j^{-z}$$ and we introduce two real functions $$f_{n}(c)$$, $$g_{n}(c)$$, $$c\in \mathbb {R}$$, associated with the end-points of the interval of variation of the variable x of the analytic variety $$| \zeta _{n}^{*}(z)| =p_{k_{n}}^{-c}$$, where $$\zeta _{n}^{*}(z):=\zeta _{n}(z)-p_{k_{n}}^{-z}$$ and $$p_{k_{n}}$$ is the last prime not exceeding n. The analysis of fixed point properties of $$f_{n}$$, $$g_{n}$$ and the behavior of such functions allow us to explain the distribution of the real parts of the zeros of $$\zeta _{n}(z)$$. Furthermore, the fixed points of $$f_{n}$$, $$g_{n}$$ characterize the set $$\mathscr {P}^{*}$$ of prime numbers greater than 2 and the set $$\mathscr {C}^{*}$$ of composite numbers greater than 2, proving in this way how close those functions from Arithmetic are. Finally, from the study of the graphs of $$f_{n}$$, $$g_{n}$$ we deduce important properties about the set $$R_{\zeta _{n}(z)}:=\overline{\{ \mathfrak {R}z:\zeta _{n}(z)=0\} }$$ and the bounds $$a_{\zeta _{n}(z)}:=\inf \{\mathfrak {R}z:\zeta _{n}(z)=0\}$$, $$b_{\zeta _{n}(z)}:=\sup \{ \mathfrak {R}z:\zeta _{n}(z)=0\}$$ that define the critical strip $$[ a_{\zeta _{n}(z)},b_{\zeta _{n}(z)}] \times \mathbb {R}$$ where are located all the zeros of $$\zeta _{n}(z)$$.
Gaspar Mora

### On Lindelöf -Spaces

In Honour of Manuel López-Pellicer
Abstract
We revisit the notion of Lindelöf $$\varSigma$$-space giving a general overview about this question. For that, we deal with the Lindelöf property to introduce Lindelöf $$\varSigma$$-spaces in order to make a description of the “goodness” of such a type of spaces, making special emphasis in the duality between X and $$C_p(X)$$ respect to some topological properties, more specifically, topological properties in which different cardinal functions are involved. Classical results are linked with more recent results.
María Muñoz-Guillermo

### Some Extensions of the Class of -Spaces

In Honour of Manuel López-Pellicer
Abstract
The $$\ell _p$$ spaces play a central role in the classical theory of operators and finitely generated tensor norms. In this setting Lindenstrauss and Pelczyǹski (Stud Math 29: 275–326, 1968) introduced the class of the $$\mathscr {L}^p$$-spaces in 1968. By construction the class $$\mathscr {L}^p$$ places us squarely in the so called Local Theory, that deals with the study of Banach spaces through or in terms of their finite-dimensional subspaces, see Pietsch (Stud Math 135: 273–298, 1999). Although the definition of the spaces of the class $$\mathscr {L}^p$$ from the $$\ell _p$$ spaces is finite-dimensional, it has many implications in the global structure of the $$\mathscr {L}^p$$-spaces. And it is clear that, in general, the nice behavior of the $$\mathscr {L}^p$$ spaces is a consequence of some good properties of the $$\ell _p$$ spaces, which make them a basic pillar of the Banach spaces theory. Anyway, if we replace $$\ell _p$$ for another sequence space $$\lambda$$, one expects that some important new problems will appear, since $$\ell _p$$ and hence $$\mathscr {L}^p$$ seem to be more or less irreplaceable. Our purpose is to present some conclusions we reached in this way in relation to the study of certain concrete tensor norms defined through different sequence spaces, making a small chronological journey, step by step, attempt after attempt. This study has an earlier version entitled “On the classes of $$\mathscr {L}^{\lambda }$$, quasi-$$\mathscr {L}^{\lambda }$$ and $$\mathscr {L}^{\lambda ,g}$$ spaces”, published in the Proceedings of the American Mathematical Society (Rivera in Proc Amer Math Soc 133: 2035–2044, 2005).
María José Rivera
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