Cleon S. Barroso
We establish an approximate fixed point result for self-maps on compact convex subsets of Hausdorff topological vector spaces where continuity is not a necessary condition.
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We give conditions for boundedness of Hausdorff operators on real Hardy spaces $H^1$ over homogeneous spaces of locally compact groups with local doubling property. The special case of the hyperbolic plane is considered.
The aim of this note is to give the boundedness conditions for Hausdorff operators on Hardy spaces $H^{1}$ with the norm defined via $(1,q)$ atoms over homogeneous spaces of Lie groups with doubling property and to apply results we obtain to generalized Delsarte operators and to Hausdorff operators over multidimensional spheres.
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.
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.
We characterize those compact Hausdorff spaces K such that the proyective tensor product of C(K) by itself has the Dunford-Pettis Property, answering thus in the negative a question posed by Castillo and Gonzalez.
Let $A$ be the disk algebra, $Ω$ be a compact Hausdorff space and $μ$ be a finite Borel measure in $Ω$. It is shown that the dual of $C(Ω,A)$ has the Dunford-Pettis Property. This proved in particular that the spaces $C(Ω,A)$ and $L^1(μ,A*)$ have the Dunford-Pettis property.
We show that the pair $(C(K),X)$ has the Bishop-Phelps-Bollobaás property for operators if $K$ is a compact Hausdorff space and $X$ is a uniformly convex space.
Let $E$ be a separable Banach space and $Ω$ be a compact Hausdorff space. It is shown that the space $C(Ω,E)$ has property (V) if and only if $E$ does. Similar result is also given for Bochner spaces $L^p(μ,E)$ if $1<p<\infty$ and $μ$ is a finite Borel measure on $Ω$.
It is shown that the Bishop-Phelps-Bollobás theorem holds for bilinear forms on the complex $C_0(L_1)\times C_0(L_2)$ for arbitrary locally compact topological Hausdorff spaces $L_1$ and $L_2$.
For a compact Hausdorff space $S$, we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on $S$, denoted by $C(S)$, satisfies property-$(P_1)$ (the set-valued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of $C(S)$. Various stability results related to property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps are also established. As a consequence, we derive that if $Y$ is a proximinal finite co-dimensional subspace of $c_0$ then the closed unit ball of $Y$ sat…
Tingley's problem for complex Banach spaces which do not satisfy the Hausdorff distance condition
针对不满足 Hatori 充分条件的一类复 Banach 空间，本文证明其单位球面满等距仍可延拓为全空间满实线性等距，并以 Lip([0,1]) 和 C^1([0,1]) 为例。
We introduce a new technique for the study of the local extension property (LEP) for boolean algebras and we use it to show that the clopen algebra of every compact Hausdorff space $K$ of finite height has LEP. This implies, under appropriate additional assumptions on $K$ and Martin's Axiom, that every twisted sum of $c_0$ and $C(K)$ is trivial, generalizing a recent result by Marciszewski and Plebanek.
We investigate the construction of a Hausdorff uo-Lebesgue topology on a vector lattice from a Hausdorff (o)-Lebesgue topology on an order dense ideal, and what the properties of the topologies thus obtained are. When the vector lattice has an order dense ideal with a separating order continuous dual, it is always possible to supply it with such a topology in this fashion, and the restriction of this topology to a regular sublattice is then also a Hausdorff uo-Lebesgue topology. A regular vector sublattice of $\mathrm{L}_0(X,Σ,μ)$ for a semi-finite measure $μ$ falls into this category, and the…
We provide a version for operators of the Bishop-Phelps-Bollobás Theorem when the domain space is the complex space $C_0(L)$. In fact we prove that the pair $(C_0(L), Y)$ satisfies the Bishop-Phelps-Bollobás property for operators for every Hausdorff locally compact space $L$ and any $\mathbb{C}$-uniformly convex space. As a consequence, this holds for $Y= L_p (μ)$ ($1 \le p < \infty $).
We study $C$-rich spaces, lush spaces, and $C$-extremely regular spaces concerning with the Mazur-Ulam property. We show that a uniform algebra and the real part of a uniform algebra with the supremum norm are $C$-rich spaces, hence lush spaces. We prove that a uniformly closed subalgebra of the algebra of complex-valued continuous functions on a locally compact Hausdorff space which vanish at infinity is $C$-extremely regular provided that it separates the points of the underlying space and has no common zeros. In section 3 we exhibit descriptions on the Choquet bounday, the \vSilov bounday, …
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 …
A Note on Capacity Muckenhoupt Weights with respect to the Hausdorff Content
本文证明在固定指标 p 时，Capacity Muckenhoupt 权类关于指标 δ 不满足自改进性质，补充了该类权性质的研究。
We obtain a refinement of a selection principle for $(\mathcal{K}, λ)$-wide-$(s)$ sequences in Banach spaces due to Rosenthal. This result is then used to show that if $C$ is a bounded, non-weakly compact, closed convex subset of a Banach space $X$, then there exists a Hausdorff vector topology $τ$ on $X$ which is weaker than the weak topology, a closed, convex $τ$-compact subset $K$ of $C$ and an affine bi-Lipschitz map $T: K\to K$ without fixed points.
A subspace $X$ of a Banach space $Y$ has $\textit{Property U}$ whenever every continuous linear functional on $X$ has a unique norm-preserving (i.e., Hahn$-$Banach) extension to $Y$ (Phelps, 1960). Throughout this document we introduce and develop a systematic study of the existence of $\textit{U-embeddings}$ between Banach spaces $X$ and $Y$, that is, isometric embeddings of $X$ into $Y$ whose ranges have property U. In particular, we are interested in the case that $Y=C(K)$, where $K$ is a compact Hausdorff topological space. We provide results for general Banach spaces and for some specific…
In this paper, we define concept of approximate fixed point property of a function and a set in intuitionistic fuzzy normed space. Furthermore, we give intuitionistic fuzzy version of some class of maps used in fixed point theory and investigate approximate fixed point property of these maps.
Let $X$ be a Hausdorff compact space and $C(X)$ be the algebra of all continuous complex-valued functions on $X$, endowed with the supremum norm. We say that $C(X)$ is (approximately) $n$-th root closed if any function from $C(X)$ is (approximately) equal to the $n$-th power of another function. We characterize the approximate $n$-th root closedness of $C(X)$ in terms of $n$-divisibility of first $\check {\rm C}$ech cohomology groups of closed subsets of $X$. Next, for each positive integer $m$ we construct $m$-dimensional metrizable compactum $X$ such that $C(X)$ is approximately $n$-th root …
C(X) denotes the space of continuous complex-valued functions on the compact Hausdorff space X. X has the CSWP if every subalgebra of C(X) which separates points and contains the constant functions is dense in C(X). W. Rudin showed that all scattered X have the CSWP. We describe a class of non-scattered X with the CSWP; by another result of Rudin, such X cannot be metrizable.
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 …
Let $M$ be a compact Hausdorff space. We prove that in this paper, every self--adjoint matrix over $C(M)$ is approximately diagonalizable iff $\dim M\le 2$ and $\HO^2(M,\mathbb Z)\cong 0$. Using this result, we show that every unitary matrix over $C(M)$ is approximately diagonalizable iff $\dim M\le 2$, $\HO^1(M,\mathbb Z)\cong\HO^2(M,\mathbb Z)\cong 0$ when $M$ is a compact metric space.
Let G be a compact group acting on a Banach space L by means of linear isometries. The action of G on L induces a natural continuous action on cc(L), the hyperspace of all compact convex subsets of L endowed with the Hausdorff metric topology. The main result of this paper states that the G-space cc(L) is a G-AE. Under some extra assumptions, this result can be extended to CB(L), the hyperspace of all closed and bounded convex subsets of L.
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