Nihat G. Gogus, Tony L. Perkins, Evgeny A. Poletsky
Edwards' Theorem establishes duality between a convex cone in the space of continuous functions on a compact space and the set of representing or Jensen measures for this cone. In this paper we prove non-compact versions of this theorem.
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Let $G$ be a locally compact group and $ω$ be a continuous weight on $G$. In this paper, we first characterize bicontinuous biseparating algebra isomorphisms between weighted $L^p$-algebras. As a result we extend previous results of Edwards, Strichartz and Parrot on algebra isomorphisms between $L^p$-algebras to the setting of weighted $L^p$-algebras. We then study the automorphisms of certain weighted $L^p$-algebras on integers, applying known results on composition operators on classical function spaces.
It has been very recently discovered that there are compact linear operators between Banach spaces which cannot be approximated by norm attaining operators. The aim of this expository paper is to give an overview of those examples and also of sufficient conditions ensuring that compact linear operators can be approximated by norm attaining operators. To do so, we introduce the analogues for compact operators of Lindenstrauss properties A and B.
Locally $L^0$-convex modules were introduced in [D. Filipovic, M. Kupper, N. Vogelpoth. Separation and duality in locally $L^0$-convex modules. J. Funct. Anal. 256(12), 3996-4029 (2009)] as the analytic basis for the study of conditional risk measures. Later, the algebra of conditional sets was introduced in [S. Drapeau, A. Jamneshan, M. Karliczek, M. Kupper. The algebra of conditional sets and the concepts of conditional topology and compactness. J. Math. Anal. Appl. 437(1), 561-589 (2016)]. In this paper we study locally $L^0$-convex modules, and find exactly which subclass of locally $L^0$-…
We introduce two measures of weak non-compactness $Ja_E$ and $Ja$ that quantify, via distances, the idea of boundary behind James' compactness theorem. These measures tell us, for a bounded subset $C$ of a Banach space $E$ and for given $x^*\in E^*$, how far from $E$ or $C$ one needs to go to find $x^{**}\in \bar{C}^{w^*}\subset E^{**}$ with $x^{**}(x^*)=\sup x^* (C)$. A quantitative version of James' compactness theorem is proved using $Ja_E$ and $Ja$, and in particular it yields the following result: {\it Let $C$ be a closed convex bounded subset of a Banach space $E$ and $r>0$. If there is …
In this paper we prove Korovkin type theorems for sequences of sublinear, monotone and weak additive operators acting on function spaces C(X); where X is a compact or a locally compact metric space. Our results are illustrated by a series of examples
We construct the generalized Lebesgue--Bochner spaces $L^p(μ,\varPi)$ for positive measures $μ$ and for suitable real or complex topological vector spaces $\varPi$ so that for $1<p<+\infty$ and Banachable $\varPi$ with separable topology the strong dual of the classical Bochner space $L^p(μ,\varPi)$ becomes canonically represented by $L^{p^*}(μ,\varPi_σ')\,$. Hence we need no separability assumption of the norm topology of the strong dual $\varPi_β'$ of $\varPi$. For $p=1$ and for suitably restricted positive measures $μ$ we even get a similar result without any separability of the norm topolo…
Multiresolution Analysis for Compactly Supported Interpolating Tensor Product Wavelets (long version)
本研究运用巴拿赫空间张量积理论，为两类n维函数空间构建插值张量积MRA及其对偶MRA，并将Besov空间范数等价结果推广到n维情形。
In this short note we give a proof of the refined version of the uniform invariant approximation property for compact (non-commutative) groups following the Bourgain's approach.
Chapter 12: The moment problem on compact semi-algebraic sets (revised version)
本文是著作第12章的修订版，为二次模和半环的Archimedean Positivstellensätze提出统一方法，并补充了半环Positivstellensätze与矩问题的新结果。
A Banach space $X$ has Pełczy\' nski's property (V) if for every Banach space $Y$ every unconditionally converging operator $T\colon X\to Y$ is weakly compact. In 1962, Aleksander Pełczy\' nski showed that $C(K)$ spaces for a compact Hausdorff space $K$ enjoy the property (V), and some generalizations of this theorem have been proved since then. We introduce several possibilities of quantifying the property (V). We prove some characterizations of the introduced quantitative versions of this property, which allow us to prove a quantitative version of Pelczynski's result about $C(K)$ spaces and …
A positive version of Miljutin's Theorem
本文证明了Miljutin定理的正版本：对任意不可数紧度量空间K和L，存在正同构T:C(K)→C(L)，其逆为正则算子，且范数乘积有界，并推广至非可度量化情形及无限紧群。
A version of Arzelà-Ascoli theorem for $X$ being $σ$-locally compact Hausdorff space is proved. The result is used in proving compactness of Fredholm, Hammerstein and Urysohn operators. Two fixed point theorems, for Hammerstein and Urysohn operator, are derived on the basis of Schauder theorem.
In this paper we study various convolution-type algebras associated with a locally compact quantum group from cohomological and geometrical points of view. The quantum group duality endows the space of trace class operators over a locally compact quantum group with two products which are operator versions of convolution and pointwise multiplication, respectively; we investigate the relation between these two products, and derive a formula linking them. Furthermore, we define some canonical module structures on these convolution algebras, and prove that certain topological properties of a quant…
We give a necessary and sufficient condition on a compact semitopological quantum semigroup which turns it into a compact quantum group. In particular, we obtain a generalisation of Ellis's joint continuity theorem. We also investigate the question of the existence of the Haar state on a compact semitopological quantum semigroup and prove a "noncommutative" version of the converse Haar's theorem.
Continuous version of the Choquet Integral Reperesentation Theorem
Choquet - Bishop - de Leeuw定理指出，局部凸拓扑T2空间的紧凸子集的每个元素都是该集合的极值点集支持的概率测度的重心。根据Edgar - Mankiewicz的结果，这对于非空闭有界凸集也成立，只要它具有Radon - Nikodym性质.在该文件中，它表明，Choquet型定理也适用于“移动”集：它们是某个多功能的值。也就是说，存在一个合适的弱 * 连续的家庭的概率测度“几乎代表”这样的集合的点被证明。紧和非紧的情况下被认为是。作为推论，得到了Krein-Milman定理的连续形式.
When applying the classical Stone-Weierstrass common version in Probability Theory for example, and in other fields as well, problems may arise if all points of the compact set are not separated. A solution may consist in going back to the proof and finding alternative versions. In this note, we did it and come back with two flexible versions which are easily used for the needs of classical Probability Theory.
We introduce a so-called restricted, in particular, discrete version of (Banach) Grand Lebesgue Spaces (GLS), investigate its properties and derive the conditions of coincidence with the classical ones. We show also that these spaces forms also a Banach algebra relative the convolution operation on the unimodular local compact topological group equipped with Haar's measure, alike in the complete GLS case.
Heisenberg uncertainty inequalities for locally compact abelian groups
本文针对局部紧阿贝尔群证明了海森堡不确定性原理，比较了欧几里得情况下的下界，并给出了局部域上的不确定性原理。
In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calderón-Zygmund operators, bilinear fractional integrals and bilinear Fouri…
A Banach space X has Pelczynski's property (V) if for every Banach space Y every unconditionally converging operator T: X -> Y is weakly compact. H. Pfitzner proved that C*-algebras have Pelczynski's property (V). In the preprint "H. Krulisova: Quantification of Pelczynski's property (V)" the author explores possible quantifications of the property (V) and shows that C(K) spaces for a compact Hausdorff space K enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner's theorem. Moreover, we prove that in dual Banach spaces a quantitative …
We present a connected version of the compact $L$-space constructed by Kenneth Kunen under CH. We show that this provides a Corson compact space $K$ such that the Banach space $C(K)$ is isomorphic to no space of continuous function on a zero-dimensional compactum.
Let $E$ be a Banach space such that $E'$ has the Radon-Nikodým property. The aim of this work is to connect relative weak compactness in the $E$-valued martingale Hardy space $H^{1}(μ,E)$ to a convex compactness criterion in a weaker topology, such as the topology of uniform convergence on compacts in measure. These results represent a dynamic version of the deep result of Diestel, Ruess, and Schachermayer on relative weak compactness in $L^{1}(μ,E)$. In the reflexive case, we obtain a Kadec-Pełczyński dichotomy for $H^{1}(μ,E)$-bounded sequences, which decomposes a subsequence into a relative…
The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in lieu of compactness in a variety of cases. Specifically, we establish convex compactness for certain familiar classes of subsets of the set of positive random variables under the topology induced by convergence in probability. Two applications in infinite-dimensional optimization - attainment of infima and a version of the Minimax theorem - are given. Moreove…
Theorems about characterization of finite rank Toeplitz operators in Fock-Segal-Bargmann spaces, known previously only for symbols with compact support, are carried over to symbols without that restriction, however with a rather rapid decay at infinity. The proof is based upon a new version of the Stone-Weierstrass approximation theorem.
A Tropical Version of the Schauder Fixed Point Theorem
证明了热带线性空间的紧子集的Schauder不动点定理的热带形式。
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