M. L. Lourenço, V. C. C. Miranda
In this paper, we study some geometric properties in Banach lattices and the class of almost limited completely continuous operators. For example, we study Banach lattices with the property (d) and we give a new characterization of this property concerning the solid hull of almost limited sets.
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Using the notion of complete compactness introduced by H. Saar, we define completely almost periodic functionals on completely contractive Banach algebras. We show that, if $(M,Γ)$ is a Hopf--von Neumann algebra with $M$ injective, then the space of completely almost periodic functionals on $M_\ast$ is a $C^\ast$-subalgebra of $M$.
Let $G$ be a metrizable, compact abelian group and let $Λ$ be a subset of its dual group $\hat G$. We show that $C_Λ(G)$ has the almost Daugavet property if and only if $Λ$ is an infinite set, and that $L^1_Λ(G)$ has the almost Daugavet property if and only if $Λ$ is not a $Λ(1)$ set.
We study almost squareness and the strong diameter two property in the setting of projective (symmetric) tensor product. We prove that almost squareness is preserved by taking projective tensor product, providing non-trivial examples of ASQ projective tensor product spaces. Furthermore, we give sufficient conditions for a projective symmetric tensor product to have the strong diameter two property which extend most of the known results and provide new examples of such spaces with the strong diameter two property.
We study the almost Daugavet property, a generalization of the Daugavet property. It is analysed what kind of subspaces and sums of Banach spaces with the almost Daugavet property have this property as well. The main result of the paper is: if $Z$ is a closed subspace of a separable almost Daugavet space $X$ such that the quotient space $X/Z$ contains no copy of $\ell_1$, then $Z$ has the almost Daugavet property, too.
We introduce the (T)-property, and prove that every Banach space with the (T)-property has the Mazur-Ulam property (briefly MUP). As its immediate applications, we obtain that almost-CL-spaces admitting a smooth point(specially, separable almost-CL-spaces) and a two-dimensional space whose unit sphere is a hexagon has the MUP. Furthermore, we discuss the stability of the spaces having the MUP by the $c_0$- and $\ell_1$-sums, and show that the space $C(K,X)$ of the vector-valued continuous functions has the the MUP, where $X$ is a separable almost-CL-space and $K$ is a compact metric space.
It is proved that the unital Banach algebra of almost periodic functions of several variables with Bohr-Fourier spectrum in a given additive semigroup is an Hermite ring. The same property holds for the Wiener algebra of functions that in addition have absolutely convergent Bohr-Fourier series. As applications of the Hermite property of these algebras, we study factorizations of Wiener--Hopf type of rectangular matrix functions and the Toeplitz corona problem in the context of almost periodic functions of several variables.
In the article is shown that classes of finite equivalence that are generated by Lebesgue-Riesz spaces Lp have the amalgamation property if and only if p is not equal to an even natural number that is strongly large then 2.
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…
Let $(M,Γ)$ be a Hopf--von Neumann algebra, so that $M_\ast$ is a completely contractive Banach algebra. We investigate whether the product of two elements of $M$ that are both weakly almost periodic functionals on $M_\ast$ is again weakly almost periodic. For that purpose, we establish the following factorization result: If $M$ and $N$ are injective von Neumann algebras, and if $x, y \in M \bar{\otimes} N$ correspond to weakly compact operators from $M_\ast$ to $N$ factoring through reflexive operator spaces $X$ and $Y$, respectively, then the operator corresponding to $xy$ factors through th…
In this paper, we first use the Bohr property to give a definition of Besicovitch almost periodic functions, and study some basic properties of Besicovitch almost periodic functions, including the equivalence of the Bohr property and the Bochner property. Then, as an application, we use the contraction principal to obtain the existence and uniqueness of Besicovitch almost periodic solutions for a class of abstract semi-linear differential equations with delay.
We study those Banach spaces $X$ for which $S_X$ does not admit a finite $\eps$-net consisting of elements of $S_X$ for any $\eps < 2$. We give characterisations of this class of spaces in terms of $\ell_1$-type sequences and in terms of the almost Daugavet property. The main result of the paper is: a separable Banach space $X$ is isomorphic to a space from this class if and only if $X$ contains an isomorphic copy of $\ell_1$.
In the present work we give some sufficient conditions to obtain a unique almost automorphic solution to abstract nonlinear integral equations which are simultaneously of advanced and delayed type and also a unique asymptotically almost automorphic mild solution to abstract integro-differential equations with nonlocal initial conditions, both situations are posed on Banach spaces. Also, we develop a Bohr-Neugebauer type result for the abstract integral equations. Before that, we introduce the notion of $λ$-bounded functions, develop the appropriate abstract theory and discuss the almost period…
Although it has been shown that, from the prevalence point of view, the elements of the S^ νspaces are almost surely multifractal, we show here that they also almost surely satisfy a weak uniform irregularity property.
We study the closure of the unitary orbit of a given point in the non-commutative Choquet boundary of a unital operator space with respect to the topology of pointwise norm convergence. This may be described more extensively as the $\ast$-representations of the $\mathrm{C}^{\ast}$-envelope that are approximately unitarily equivalent to one that possesses the unique extension property. Although these $\ast$-representations do not necessarily have the unique extension property themselves, we show that their unital completely positive extensions display significant restrictions. When the underlyi…
Partially greedy bases in Banach spaces were introduced by Dilworth et al. as a strictly weaker notion than the (almost) greedy bases. In this paper, we study two natural ways to strengthen the definition of partial greediness. The first way produces what we call the consecutive almost greedy property, which turns out to be equivalent to the almost greedy property. Meanwhile, the second way reproduces the PG property for Schauder bases but a strictly stronger property for general bases.
Interpolation and Fatou-Zygmund property for completely Sidon subsets of discrete groups (New title: Completely Sidon sets in discrete groups)
本文研究了离散群中的完全Sidon集，证明了其在有限并集下的稳定性，强调了插值性质，并给出了Fatou-Zygmund性质及闭张成空间完全同构的刻画。
Weighted shifts on directed trees are a decade old generalisation of classical shift operators in the sequence space $\ell^2$. In this paper we introduce the joint backward extension property (JBEP) for classes of weighted shifts on directed trees. If a class satisfies JBEP, the existence of a common backward extension within the class for a family of weighted shifts on rooted directed trees does not depend on the additional structure of the big tree (of fixed depth). We decide whether several classes of operators have JBEP. For subnormal or power hyponormal weighted shifts the property is sat…
A Banach space is locally almost square if, for every $y$ in its unit sphere, there exists a sequence $(x_n)$ in its unit sphere such that $\lim\|y\pm x_n\|=1$. A Banach space is weakly almost square if, in addition, we require the sequence $(x_n)$ to be weakly null. It is known that these two properties are distinct, so we aim to investigate if local almost squareness implies a weaker version of the latter property by replacing the sequence with a net. In order to achieve this result, we restrict ourselves to Banach lattices and introduce a strengthening of local almost squareness by requirin…
Let $X$ and $Y$ be Banach spaces, and $T:X^*\to Y$ be an operator. We prove that if $X$ is Asplund and $Y$ has the approximation property, then for each Radon probability $μ$ on $(B_{X^*},w^*)$ there is a sequence of $w^*$-to-norm continuous operators $T_n:X^*\to Y$ such that $\|T_n(x^*)-T(x^*)\| \to 0$ for $μ$-a.e. $x^*\in B_{X^*}$; if $Y$ has the $λ$-bounded approximation property for some $λ\geq 1$, then the sequence can be chosen in such a way that $\|T_n\|\leq λ\|T\|$ for all $n\in \mathbb{N}$. The same conclusions hold if $X$ contains no subspace isomorphic to $\ell_1$, $Y$ has the appro…
In this article, the class of all Dunford-Pettis $ p $-convergent operators and $ p $-Dunford-Pettis relatively compact property on Banach spaces are investigated. Moreover, we give some conditions on Banach spaces $ X $ and $ Y $ such that the class of bounded linear operators from $ X$ to $ Y $ and some its subspaces have the $ p $-Dunford-Pettis relatively compact property. In addition, if $ Ω$ is a compact Hausdorff space, then we prove that dominated operators from the space of all continuous functions from $ K $ to Banach space $ X $ (in short $ C(Ω,X) $) taking values in a Banach space …
Best approximations in metric spaces with property strongly UC
本文引入强UC性质，证明度量空间中几乎循环ψ-压缩映射最佳逼近的存在性，并推广了已有结果至自反Banach空间。
A theorem of Davis, Figiel, Johnson and Pełczyński tells us that weakly-compact operators between Banach spaces factor through reflexive Banach spaces. The machinery underlying this result is that of the real interpolation method, which has been adapted to the category of operator spaces by Xu, showing the this factorisation result also holds for completely bounded weakly-compact maps. In this note, we show that Xu's ideas can be adapted to give an intrinsic characterisation of when a completely contractive Banach algebra arises as a closed subalgebra of the algebra of completely bounded opera…
We prove that (almost) universal disposition and separable universal disposition of Banach spaces are not $3$-space properties.
In contrast with the separable case, we prove that the existence of almost $L$-orthogonal vectors in a nonseparable Banach space $X$ (octahedrality) does not imply the existence of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$, which shows that the answer to an environment question in [9] is negative. Furthermore, we prove that the abundance of almost $L$-orthogonal vectors in a Banach space $X$ (almost Daugavet property) whose density character is $ω_1$ implies the abundance of nonzero vectors in $X^{**}$ being $L$-orthogonal to $X$. In fact, we get that a Banach space $X$ whose den…
In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space which is a vectorial generalization of a result of Chalendar-Gallardo-Partington (C-G-P). Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space.
A Banach space $\X$ has the complete continuity property (CCP) if each bounded linear operator from $L_1$ into $\X$ is completely continuous (i.e., maps weakly convergent sequences to norm convergent sequences). The main theorem shows that a Banach space failing the CCP (resp., failing the CCP and failing cotype) has a subspace with a finite dimensional decomposition (resp., basis) which fails the CCP.
We prove that if $G$ is a discrete group and $(A,G,α)$ is a C*-dynamical system such that the reduced crossed product $A\rtimes_{r,α} G$ possesses property (SOAP) then every completely compact Herz-Schur $(A,G,α)$-multiplier can be approximated in the completely bounded norm by Herz-Schur $(A,G,α)$-multipliers of finite rank. As a consequence, if $G$ has the approximation property (AP) then the completely compact Herz-Schur multipliers of $A(G)$ coincide with the closure of $A(G)$ in the completely bounded multiplier norm. We study the class of invariant completely compact Herz-Schur multiplie…
In 2010, Lafforgue and de la Salle gave examples of noncommutative Lp-spaces without the operator space approximation property (OAP) and, hence, without the completely bounded approximation property (CBAP). To this purpose, they introduced the property of completely bounded approximation by Schur multipliers on Sp and proved that for p < 4/3 and p > 4 the groups SL(n,Z), with n \geq 3, do not have it. Since for 1 < p < \infty the property of completely bounded approximation by Schur multipliers on Sp is weaker than the approximation property of Haagerup and Kraus (AP), these groups were also t…
Certain operator algebras A on a Hilbert space have the property that every densely defined linear transformation commuting with A is closable. Such algebras are said to have the closability property. They are important in the study of the transitive algebra problem. More precisely, if A is a two-transitive algebra with the closability property, then A is dense in the algebra of all bounded operators, in the weak operator topology. In this paper we focus on algebras generated by a completely nonunitary contraction, and produce several new classes of algebras with the closability property. We s…
The Aaronson-Ambainis conjecture (Theory of Computing '14) says that every low-degree bounded polynomial on the Boolean hypercube has an influential variable. This conjecture, if true, would imply that the acceptance probability of every $d$-query quantum algorithm can be well-approximated almost everywhere (i.e., on almost all inputs) by a $\mathrm{poly}(d)$-query classical algorithm. We prove a special case of the conjecture: in every completely bounded degree-$d$ block-multilinear form with constant variance, there always exists a variable with influence at least $1/\mathrm{poly}(d)$. In a …
We present a measure-theoretic condition for a property to hold ``almost everywhere'' on an infinite-dimensional vector space, with particular emphasis on function spaces such as $C^k$ and $L^p$. Like the concept of ``Lebesgue almost every'' on finite-dimensional spaces, our notion of ``prevalence'' is translation invariant. Instead of using a specific measure on the entire space, we define prevalence in terms of the class of all probability measures with compact support. Prevalence is a more appropriate condition than the topological concepts of ``open and dense'' or ``generic'' when one desi…
Recently, a complete characterization of connected Lie groups with the Approximation Property was given. The proof used of the newly introduced property (T*). We present here a short proof of the same result avoiding the use of property (T*). Using property (T*), however, the characterization is extended to every almost connected group. We end with some remarks about the impossibility of going beyond the almost connected case.
Akcoglu and Suchaston proved the following result: Let $T:L^1(X,{\cf},\m)\to L^1(X,{\cf},\m)$ be a positive contraction. Assume that for $z\in L^1(X,{\cf},\m)$ the sequence $(T^nz)$ converges weakly in $L^1(X,{\cf},\m)$, then either $\lim\limits_{n\to\infty}\|T^nz\|=0$ or there exists a positive function $h\in L^1(X,{\cf},\m)$, $h\neq 0$ such that $Th=h$. In the paper we prove an extension of this result in finite von Neumann algebra setting, and as a consequence we obtain that if a positive contraction of a noncommutative $L^1$-space has no non zero positive invariant element, then its mixing…
We prove that the projection on a completely entangled subspace S of maximum dimension in a multipartite quantum system obtained by Parthasarathy is not positive under partial transpose. We next show that several positive operators with range in S also have the same property. In this process we construct an orthonormal basis of S and provide a linking theorem to link the constructions of completely entangled subspaces due to Parthasarthy, Bhat and Johnston.
A Separable Hilbertian Operator Space with CBAP and No Completely Bounded Basis
本文构造了一个具有1-CBAP但无完全有界基的可分希尔伯特算子空间，表明完全有界框架不必产生完全有界基，并利用Oikhberg定理完成构造。
The aim of generalized phase retrieval is to recover $\mathbf{x}\in \mathbb{F}^d$ from the quadratic measurements $\mathbf{x}^*A_1\mathbf{x},\ldots,\mathbf{x}^*A_N\mathbf{x}$, where $A_j\in \mathbf{H}_d(\mathbb{F})$ and $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. In this paper, we study the matrix set $\mathcal{A}=(A_j)_{j=1}^N$ which has the almost everywhere phase retrieval property. For the case $\mathbb{F}=\mathbb{R}$, we show that $N\geq d+1$ generic matrices with prescribed ranks have almost everywhere phase retrieval property. We also extend this result to the case where $A_1,\ldots,A_N$ a…
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