George A. Willis
It is shown how to construct, given a Banach space which does not have the approximation property, another Banach space which does not have the approximation property but which does have the compact approximation property.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Two new Banach space moduli, that involve weak convergent sequences, are introduced. It is shown that if either one of these moduli are strictly less than 1 then the Banach space has Property($K$)
Sequential cone-compactness does not imply cone-compactness
针对文献[1]提出的问题，本文构造反例说明在无可分性条件时序列锥紧性不蕴含锥紧性。
We prove that a Banach space $E$ has the compact range property (CRP) if and only if for any given $C^*$-algebra $\cal A$, every absolutely summing operator from $\cal A$ into $E$ is compact.
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.
We show that Sobczyk's Theorem holds for a new class of Banach spaces, namely spaces of continuous functions on linearly ordered compacta.
A short note on hereditary Mazur intersection property
本文通过Borwein和Fabian的例子证明继承性Mazur交集性质不蕴含弗雷歇平滑性。
We study the Bishop-Phelps-Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that $C_0(L)$ spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space $L$. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of $C_0(L)$ spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of $\…
We present a characterization of the continuous increasing surjections $ϕ:K\to L$ between compact lines $K$ and $L$ for which the corresponding subalgebra $ϕ^*C(L)$ has the $c_0$-extension property in $C(K)$. A natural question arising in connection with this characterization is shown to be independent of the axioms of ZFC.
Let $X$ be a Banach space and let $C$ be a closed convex bounded subset of $X$. It is proved that $C$ is weakly compact if, and only if, $C$ has the {it generic} fixed point property ($\mathcal{G}$-FPP) for the class of $L$-bi-Lipschitz affine mappings for every $L>1$. It is also proved that if $X$ has Pełczyński's property $(u)$, then either $C$ is weakly compact, contains an $\ell_1$-sequence or a $\mathrm{c}_0$-summing basic sequence. In this case, weak compactness of $C$ is equivalent to the $\mathcal{G}$-FPP for the strengthened class of affine mappings that are uniformly bi-Lipschitz. We…
The weak maximizing property, compact perturbations and duality
构造一族与\u2113_2同构的自反Banach空间，证明弱极大化性质不对偶稳定，且紧扰动性质不蕴含弱极大化性质。
We construct a Bourgain-Delbaen $\mathscr{L}_\infty$-space $\mathfrak{X}_{Kus}$ with strongly heterogenous structure: any bounded operator on $\mathfrak{X}_{Kus}$ is a compact perturbation of a multiple of the identity, whereas the space $\mathfrak{X}_{Kus}$ is saturated with unconditional basic sequences.
For a locally convex $^*$-algebra $A$ equipped with a fixed continuous $^*$-character $\varepsilon$, we define a cohomological property, called property $(FH)$, which is similar to character amenability. Let $C_c(G)$ be the space of continuous functions on a second countable locally compact group $G$ with compact supports, equipped with the convolution $^*$-algebra structure and a certain inductive topology. We show that $(C_c(G), \varepsilon_G)$ has property $(FH)$ if and only if $G$ has property $(T)$. On the other hand, many Banach algebras equipped with canonical characters have property $…
We prove that a closed convex subset of a Banach space is (super-)weakly compact if and only if it is (super)-ergodic. As a consequence we deduce that super weakly compact sets are characterized by the fixed point property for continuous affine mappings. We also prove that the M-(fixed point property for affine isometries) implies the Banach-Saks property.
On approximation properties related to unconditionally p-compact operators and Sinha-Karn p-compact operators
该研究在1≤p<2情形下推导了无条件p-紧算子理想的逼近性质新结论，否定回答了Kim的p=1相关问题，并证明ℓ^q的闭子空间可对所有r≥p失效Sinha-Karn r-紧逼近性质。
A hereditarily indecomposable Banach space $\mathfrak{X}_{\mathfrak{nr}}$ is constructed that is the first known example of a $\mathscr{L}_\infty$-space not containing $c_0$, $\ell_1$, or reflexive subspaces and answers a question posed by J. Bourgain. Moreover, the space $\mathfrak{X}_{\mathfrak{nr}}$ satisfies the "scalar-plus-compact" property and it is the first known space without reflexive subspaces having this property. It is constructed using the Bourgain-Delbaen method in combination with a recent version of saturation under constraints in a mixed-Tsirelson setting. As a result, the s…
Quantitative Schur property and measures of weak non-compactness
本文研究了Banach空间的定量化Schur性质，探讨其与弱非紧性测度的关系，并给出了相关结果和例子。
Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property
本文研究了具有逼近性质失败的巴拿赫空间上紧-近似可算算子商代数的结构性质，并给出了相关嵌入结果。
We study the Bishop-Phelps-Bollobás property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$ has the BPBp for compact operators, then so do $(C_0(L),Y)$ for every locally compact Hausdorff topological space $L$ and $(X,Y)$ whenever $X^*$ is isometrically isomorphic to $\ell_1$. If $X^*$ has the Radon-Nikodým property and $(\ell_1(X),Y)$ has the BPBp for compact operators, then …
We show that the space of bounded and linear operators between spaces of continuous functions on compact Hausdorff topological spaces has the Bishop-Phelps-Bollobás property. A similar result is also proved for the class of compact operators from the space of continuous functions vanishing at infinity on a locally compact and Hausdorff topological space into a uniformly convex space, and for the class of compact operators from a Banach space into a predual of an $L_1$-space.
Lipschitz free $p$-spaces for $0<p<1$ in the light of the Schur $p$-property and the compact reduction
本文引入 Schur $p$-性质及强 Schur $p$-性质，并利用适配的紧归约原理证明离散度量空间上的 Lipschitz free $p$-空间具有逼近性质，肯定回答了已有公开问题。
M-ideals of compact operators and Norm attaining operators
本文研究了紧算子的M-理想及弱最大化性质和紧扰动性质，证明了它们与性质$(M)$的关联及若干逆命题不成立，并探讨了Banach空间的几何性质。
On the stability of the Bishop's property($β$) under compact perturbations
本文证明了Banach空间上算子的Bishop性质($\beta$)、分解性质($\delta$)及可分解性在拟幂零、紧和代数可交换扰动下保持稳定，推进了局部谱理论中开放问题的研究。
In this paper, we introduce the concept of a pseudo weakly compact operator of order $ p $ between Banach spaces. Also we study the notion of $ p $-Dunford-Pettis relatively compact property which is in "general" weaker than the Dunford-Pettis relatively compact property and gives some characterizations of Banach spaces which have this property. Moreover, by using the notion of $ p $-Right subsets of a dual Banach space, we study the concepts of $ p $-sequentially Right and weak $ p $-sequentially Right properties on Banach spaces. Furthermore, we obtain some suitable conditions on Banach spac…
In this article we prove that every isometric copy of C(L) in C(K) is complemented if L is compact Hausdorff of finite height and K is a compact Hausdorff space satisfying the extension property, i.e., every closed subset of K admits an extension operator. The space C(L) can be replaced by its subspace C(L|F) consisting of functions that vanish on a closed subset F of L. We also study the class of spaces having the extension property, establishing some closure results for this class and relating it to other classes of compact spaces.
Let $X$ be a Banach space and $μ$ a probability measure. A set $K \subseteq L^1(μ,X)$ is said to be a $δ\mathcal{S}$-set if it is uniformly integrable and for every $δ>0$ there is a weakly compact set $W \subseteq X$ such that $μ(f^{-1}(W)) \geq 1-δ$ for every $f\in K$. This is a sufficient, but in general non necessary, condition for relative weak compactness in $L^1(μ,X)$. We say that $X$ has property ($δ\mathcal{S}_μ$) if every relatively weakly compact subset of $L^1(μ,X)$ is a $δ\mathcal{S}$-set. In this paper we study $δ\mathcal{S}$-sets and Banach spaces having property ($δ\mathcal{S}_μ…
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 …
For an operator ideal $\mathcal A$, we study the composition operator ideals ${\mathcal A}\circ{\mathcal K}$, ${\mathcal K}\circ{\mathcal A}$ and ${\mathcal K}\circ{\mathcal A}\circ{\mathcal K}$, where $\mathcal K$ is the ideal of compact operators. We introduce a notion of an $\mathcal A$-approximation property on a Banach space and characterise it in terms of the density of finite rank operators in ${\mathcal A}\circ{\mathcal K}$ and ${\mathcal K}\circ{\mathcal A}$. We propose the notions of $\ell_{\infty}$-extension and $\ell_{1}$-lifting properties for an operator ideal $\mathcal A$ and st…
In this paper we study conditions on a Banach space X that ensure that the Banach algebra K(X) of compact operators is amenable. We give a symmetrized approximation property of X which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct su…
The notion of $p$-compact sets arises naturally from Grothendieck's characterization of compact sets as those contained in the convex hull of a norm null sequence. The definition, due to Sinha and Karn (2002), leads to the concepts of $p$-approximation property and $p$-compact operators, which form a ideal with its ideal norm $κ_p$. This paper examines the interaction between the $p$-approximation property and the space of holomorphic functions. Here, the $p$-compact analytic functions play a crucial role. In order to understand this type of functions we define a $p$-compact radius of converge…
Compact embeddings and Pitt's property for weighted sequence spaces of Sobolev type
本文引入一类新的Sobolev型加权序列空间，证明其紧嵌入定理，并推广了非平衡统计力学中主方程谱分析的紧嵌入结果，同时证明了Pitt型定理：满足条件时，该类空间之间的线性有界变换均为紧算子。
On stability of metric spaces and Kalton's property $Q$
本文证明Kalton的Q性质蕴含度量空间的上稳定性，并通过渐近结构给出自反Banach空间上稳定性的直接证明，为粗几何与一致范畴中的空间区分提供了新工具。
A study on $\mr{F}$-simultaneous approximative $τ$-compactness property in Banach spaces
本文研究了Banach空间中F-同时近似τ-紧性性质，探讨了其与反射性、Fréchet光滑性和Kadec-Klee性质的关系。
In the article we settle down the problem of permanence of property RD under group extensions. We show that if $1\to N\to G\to Q\to 1$ is a short exact sequence of compactly generated groups such that $Q$ has property RD, and $N$ has property RD with respect to the restriction of a word-length on $G$, then $G$ has property RD. We also generalize the result of Ji and Schweitzer stating that locally compact groups with property RD are unimodular. Namely, we show that any automorphism of a locally compact group with property RD which distorts distances subexponentially, preserves the Haar measure…
Weak Minimizing Property and the Compact Perturbation Property for the Minimum Modulus
本文研究了最小模的弱极小性和紧扰动性质，证明了(c₀,c₀)不满足紧扰动性质，并展示了一个非极小达到算子在紧扰动下最小模严格增加的现象，且该现象不局限于c₀。
Let $G$ be a locally compact group and let $C^*(G)$ and $C^*_r(G)$ be the full group $C^*$-algebra and the reduced group $C^*$-algebra of $G$. We investigate the relationship between Property $(T)$ for $G$ and Property $(T)$ as well as its strong version for $C^*(G)$ and $C^*_r(G)$. We show that $G$ has Property $(T)$ if (and only if) $C^*(G)$ has Property $(T)$. In the case where $G$ is a locally compact IN-group, we prove that $G$ has Property $(T)$ if and only if $C^*_r(G)$ has strong Property $(T)$. We also show that $C^*_r(G)$ has strong Property $(T)$ for every non-amenable locally compa…
The question which led to the title of this note is the following: {\it If $X$ is a Banach space and $K$ is a compact subset of $X$, is it possible to find a compact, or even approximable, operator $v:X\to X$ such that $K\subset\ol{v(B_X)}$?} This question was first posed by P.G.Dixon [6] in connection with investigating the problem of the existence of approximate identities in certain operator algebras. We shall provide a couple of observations related to the above question and give in particular a negative answer in case of approximable operators. We shall also provide the first examples of …
The Lotz Property in Weakly Compactly Generated Spaces
研究弱紧生成Banach空间的Lotz性质，提出Calkin代数判据，结合提取定理构造出可分不可分解的Lotz空间，并证明不可分WCG空间均非Lotz。
We study $M$-ideals of compact operators by means of the property~$(M)$ introduced in \cite{Kal-M}. Our main result states for a separable Banach space $X$ that the space of compact operators on $X$ is an $M$-ideal in the space of bounded operators if (and only if) $X$ does not contain a copy of $\ell_{1}$, has the metric compact approximation property, and has property~$(M)$. The investigation of special versions of property~$(M)$ leads to results on almost isometric structure of some classes of Banach spaces. For instance, we give a simple necessary and sufficient condition for a Banach spac…
We use the notion of $\A$-compact sets, which are determined by a Banach operator ideal $\A$, to show that most classic results of certain approximation properties and several Banach operator ideals can be systematically studied under this framework. We say that a Banach space enjoys the $\A$-approximation property if the identity map is uniformly approximable on $\A$-compact sets by finite rank operators. The Grothendieck's classic approximation property is the $\K$-approximation property for $\K$ the ideal of compact operators and the $p$-approximation property is obtained as the $\mathcal N…
A locally compact group $G$ has the factorization property if the map $$C^*(G)\odot C^*(G)\ni a\otimes b\mapsto λ(a)ρ(b)\in\mathcal B(L^2(G))$$ is continuous with respect to the minimal C*-norm. This paper seeks to initiate a rigorous study of this property in the case of locally compact groups which, in contrast to the discrete case, has been relatively untouched. A partial solution to the question of when the factorization property passes to continuous embeddings is given -- a question which traces back to Kirchberg's seminal work on the topic and is known to be false in general. It is also …
Taking matrix as a synonym for a numerical function on the Cartesian product of two (in general, infinite) sets, a simple purely algebraic "reciprocity property" says that the set of rows spans a finite-dim space iff the set of columns does so. Similar topological reciprocity properties serve to define strongly compact and weakly compact matrices, featured in the well-known basic facts about almost periodic functions and about compact operators. Some properties, especially for the weak compact case, are investigated, such as the connection with the matrix having a Fubini-like property for gene…
It is known that for $σ$-compact groups Kazhdan's Property $(T)$ is equivalent to Serre's Property $(FH)$. Generalized versions of those properties, called properties $(T_{B})$ and $(F_{B})$, can be defined in terms of the isometric representations of a group on an arbitrary Banach space $B$. Property $(F_{B})$ implies $(T_{B})$. It is known that a group with Property $(T_{l_p})$ shares some properties with Kazhdan's groups, for example compact generation and compact abelianization. Moreover in the case of discrete groups, Property $(T_{l_p})$ implies Lubotzky's Property $(τ)$. In this paper w…
We prove that the Lipschitz free space over a certain type of discrete metric space has the Radon-Nikodým property. We also show that the Lipschitz free space over a complete, locally compact metric space has the Schur or approximation property whenever the Lipschitz free space over each compact subset also has this property.
In this short article, we obtained some equivalent formulations of property $T$ for a general locally compact quantum group $\mathbb{G}$, in terms of the full quantum group $C^*$-algebras $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ and the $*$-representation of $C_0^\mathrm{u}(\widehat{\mathbb{G}})$ associated with the trivial unitary corepresentation (that generalize the corresponding results for locally compact groups). Moreover, if $\mathbb{G}$ is of Kac type, we show that $\mathbb{G}$ has property $T$ if and only if every finite dimensional irreducible $*$-representation of $C_0^\mathrm{u}(\wid…
The approximation property for locally compact quantum groups
本文研究局部紧量子群的Haagerup--Kraus逼近性质，推广了前人工作，建立了多项继承性质，并引入了离散量子群的中心变体，将其与刚性C*-张量范畴的逼近性质相联系。
The relationship between the operator approximation property and the strong operator approximation property has deep significance in the theory of operator algebras. The original definitions of Effros and Ruan, unlike the classical analogues, make no mention of compact operators or compact sets. In this paper we introduce ``compact matrix sets'' which correspond to the two different operator approximation properties, and show that a space has the operator approximation property if and only if the ``operator compact'' operators are contained in the closure of the finite rank operators. We also …
Bilinear maps on C$^*$-algebras that have product property at a compact element
本文研究 C*-代数上在紧元处具有乘积性质的有界双线性映射，分别对紧 C*-代数和具有非平凡原子部分的 von Neumann 代数给出何时由该点乘积确定的判据，并应用于刻画固定点处类似同态与类似导子的映射。
We discuss a set-valued generalization of strong proximinality in Banach spaces, introduced by J. Mach [Continuity properties of Chebyshev centers, J. Approx. Theory, 29(3):223--230, 1980] as property-$(P_1)$. We establish that if the closed unit ball of a closed subspace of a Banach space $X$ possesses property-$(P_1)$ for each of the classes of closed bounded, compact and finite subsets of $X$, then so does the subspace. It is also proved that the closed unit ball of an $M$-ideal in an $L_{1}$-predual space satisfies property-$(P_{1})$ for the compact subsets of the space. For a Choquet simp…
Entropy for compact operators and results on entropy and specification
本文研究了算子的拓扑熵，证明了紧算子在Banach空间中具有有限熵，仅依赖于其特征值。同时探讨了F空间中规范性与熵的关系，指出规范性蕴含无限熵，而算子规范性蕴含正熵。
We introduce the strong Gelfand-Phillips property for locally convex spaces and give several characterizations of this property. We characterize the strong Gelfand-Phillips property among locally convex spaces admitting a stronger Banach space topology. If $C_{\mathcal T}(X)$ is a space of continuous functions on a Tychonoff space $X$, endowed with a locally convex topology $\mathcal T$ between the pointwise topology and the compact-open topology, then: (a) the space $C_{\mathcal T}(X)$ has the strong Gelfand-Phillips property iff $X$ contains a compact countable subspace $K\subseteq X$ of fin…
We prove a number of property (T) permanence results for locally compact quantum groups under exact sequences and the presence of invariant states, analogous to their classical versions. Along the way we characterize the existence of invariant weights on quantum homogeneous spaces of quotient type, and relate invariant states for LCQG actions on von Neumann algebras to invariant vectors in canonical unitary implementations, providing an application to amenability. Finally, we introduce a notion of lattice in a locally compact quantum group, noting examples provided by Drinfeld doubles of compa…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。