Jesus Araujo
For realcompact spaces X and Y we give a complete description of the linear biseparating maps between spaces of vector-valued continuous functions on X and Y, where special attention is paid to spaces of vector-valued bounded continuous functions. These results are applied to describe the linear isometries between spaces of vector-valued bounded continuous and uniformly continuous functions.
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Banach-Stone type theorems on uniformly continuous and lipschitz continuous pseudometrics
本文研究一致连续和Lipschitz连续伪度量空间中的Banach-Stone型定理，探讨其核心方法与主要贡献。
We show that if there exists a Lipschitz homeomorphism $T$ between the nets in the Banach spaces $C(X)$ and $C(Y)$ of continuous real valued functions on compact spaces $X$ and $Y$, then the spaces $X$ and $Y$ are homeomorphic provided $l(T) \times l(T^{-1})< 6/5$. By $l(T)$ and $l(T^{-1})$ we denote the Lipschitz constants of the maps $T$ and $T^{-1}$. This improves the classical result of Jarosz and the recent result of Dutrieux and Kalton where the constant obtained is 17/16. We also estimate the distance of the map $T$ from the isometry of the spaces $C(X)$ and $C(Y)$.
In this paper, we extend the Banach-Stone theorem to the non commutative case, i.e, we prove that the structure of the liminal $C^{*}$-algebras $\cal A$ determines the topology of its primitive ideal space.
One important class of tools in the study of the connections between algebraic and topological structures are the "Banach-Stone type theorems", which describe algebraic isomorphisms of algebras (or groups, lattices, etc.) of functions in terms of homeomorphisms between the underlying topological spaces. Several such theorems have been proven throughout the last century, however not all of them are comparable, and in particular no single one is the strongest. In this article, we describe a general framework which encompasses several of these results, and which allows for new applications relate…
In this paper, we extend the Banach-Stone theorem to the non commutative case, i.e, we give a partial answere to the question 2.1 of [13], and we prove that the structure of the postliminal C*-algebras A determines the topology of its primitive ideals space.
We show that under a certain topological assumption on two compact hereditary families $\F$ and $\G$ on some infinite cardinal $κ$, the corresponding combinatorial spaces $X_\F$ and $X_\G$ are isometric if and only if there is a permutation of $κ$ inducing a homeomorphism between $\F$ and $\G$. We also prove that two different regular families $\F$ and $\G$ on $ω$ cannot be permuted one to the other. Both these results strengthen the main result of \cite{BrechFerencziTcaciuc}.
Let $X$ and $Y$ be compact Hausdorff spaces, and $E$, $F$ be Banach lattices. Let $C(X,E)$ denote the Banach lattice of all continuous $E$-valued functions on $X$ equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism $\mathnormalΦ: C(X,E)\to C(Y,F)$ such that $\mathnormalΦf$ is non-vanishing on $Y$ if and only if $f$ is non-vanishing on $X$, then $X$ is homeomorphic to $Y$, and $E$ is Riesz isomorphic to $F$. In this case, $\mathnormalΦ$ can be written as a weighted composition operator: $\mathnormalΦ f(y)=\mathnormalΠ(y)(f(φ(y)))$, where $φ$ …
Pointwise semi-Lipschitz functions and Banach-Stone theorems
本文研究非对称空间上逐点半Lipschitz函数的基本性质及部分对称性的影响，并利用两个函数代数建立两个Banach-Stone型结果，其核心是一个适用于抽象函数空间的灵活拓扑型一般定理。
Let $X$ and $Y$ be completely regular spaces and $E$ and $F$ be Hausdorff topological vector spaces. We call a linear map $T$ from a subspace of $C(X,E)$ into $C(Y,F)$ a \emph{Banach-Stone map} if it has the form $Tf(y) = S_{y}(f(h(y))$ for a family of linear operators $S_{y} : E \to F$, $y \in Y$, and a function $h: Y \to X$. In this paper, we consider maps having the property: \cap^{k}_{i=1}Z(f_{i}) \neq\emptyset\iff\cap^{k}_{i=1}Z(Tf_{i}) \neq \emptyset, where $Z(f) = \{f = 0\}$. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces…
For a compact Hausdorff space $K$, we give descriptions of the dual of $C(K)^δ$, the Dedekind completion of the Banach lattice $C(K)$ of continuous, real-valued functions on $K$. We characterize those functionals which are $σ$-order continuous and order continuous, respectively, in terms of Oxtoby's category measures. This leads to a purely topological characterization of hyper-Stonean spaces.
For a space $X$ denote by $C_b(X)$ the Banach algebra of all continuous bounded scalar-valued functions on $X$ and denote by $C_0(X)$ the set of all elements in $C_b(X)$ which vanish at infinity. We prove that certain Banach subalgebras $H$ of $C_b(X)$ are isometrically isomorphic to $C_0(Y)$, for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$ is explicitly constructed as a subspace of the Stone--Čech compactification of $X$. The known construction of $Y$ enables us to examine certain properties of either $H$ or $Y$ and derive results not expected to be de…
M. Talagrand showed that, for the Cech-Stone compactification βω of the space of natural numbers, the norm and the weak topology generate different Borel structures in the Banach space C(βω). We prove that the Borel structures in C(βω) generated by the weak and the pointwise topology are also different. We also show that in C(ω*), where ω*=βω- ω, there is no countable family of pointwise Borel sets separating functions from C(ω*).
We prove a commutative Gelfand--Naimark type theorem, by showing that the set $C_s(X)$ of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space $X$ (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to $C_0(Y)$, for some unique (up to homeomorphism) locally compact Hausdorff space $Y$. The space $Y$, which we explicitly construct as a subspace of the Stone--Čech compactification of $X$, is countably compact, and if $X$ is non-separable, is moreover non-normal; in addition $C_0(Y)=C_{00}(Y)$. When the un…
Given a Boolean algebra $A$, we construct another Boolean algebra $B$ with no uncountable well-ordered chains such that the Banach space of real valued continuous functions $C(K_A)$ embeds isometrically into $C(K_B)$, where $K_A$ and $K_B$ are the Stone spaces of $A$ and $B$ respectively. As a consequence we obtain the following: If there exists an isometrically universal Banach space for the class of Banach spaces of a given uncountable density $κ$, then there is another such space which is induced by a Boolean algebra which is not universal for Boolean algebras of cardinality $κ$. Such a phe…
We prove a very general theorem concerning the estimation of the expression $\|T(\frac{a+b}{2}) - \frac{Ta+Tb}{2}\|$ for different kinds of maps $T$ satisfying some general perurbated isometry condition. It can be seen as a quantitative generalization of the classical Mazur-Ulam theorem. The estimates improve the existing ones for bi-Lipschitz maps. As a consequence we also obtain a very simple proof of the result of Gevirtz which answers the Hyers-Ulam problem and we prove a non-linear generalization of the Banach-Stone theorem which improves the results of Jarosz and more recent results of D…
The article is devoted to quasilinear operators in spaces over quaternions and octonions. Spectral theory of bounded and unbounded operators is investigated. Analogs of C^* algebras are defined and studied. Among main results are analogs of the Gelfand-Mazur, Stone, Gelfand-Naimark-Segal, von Neumann, Kaplansky, etc. theorems, but here over the skew field of quaternions and the octonion algebra.
We examine several classical concepts from topology and functional analysis, using methods of commutative algebra. We show that these various concepts are all controlled by BC R-rings and their maximal spectra. A BC R-ring is a ring A that is isomorphic to the ring of bounded continuous R-valued functions on some compact topological space X. These rings are not topologized. We prove that the category of BC R-rings is dual to the category of compact topological spaces. Next we prove that for every topological space X the ring of bounded continuous functions on it is a BC R-ring. These theorems …
We deal with isomorphic Banach-Stone type theorems for closed subspaces of vector-valued continuous functions. Let $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. For $i=1,2$, let $E_i$ be a reflexive Banach space over $\mathbb{F}$ with a certain parameter $λ(E_i)>1$, which in the real case coincides with the Schaffer constant of $E_i$, let $K_i$ be a locally compact (Hausdorff) topological space and let $\mathcal{H}_i$ be a closed subspace of $\mathcal{C}_0(K_i, E_i)$ such that each point of the Choquet boundary $\mathcal{Ch}_{\mathcal{H}_i} K_i$ of $\mathcal{H}_i$ is a weak peak point. We show that…
In this paper, we report on new results related to the existence of an adjoint for operators on separable Banach spaces and discuss a few interesting applications. (Some results are new even for Hilbert spaces.) Our first two applications provide an extension of the Poincaré inequality and the Stone-von Neumann version of the spectral theorem for a large class of $C_0$-generators of contraction semigroups on separable Banach spaces. Our third application provides a natural extension of the Schatten-class of operators to all separable Banach spaces. As a part of this program, we introduce a new…
A convex-polynomial is a convex combination of the monomials $\{1, x, x^2, \ldots\}$. This paper establishes that the convex-polynomials on $\mathbb R$ are dense in $L^p(μ)$ and weak$^*$ dense in $L^\infty(μ)$, precisely when $μ([-1,\infty)) = 0$. It is shown that the convex-polynomials are dense in $C(K)$ precisely when $K \cap [-1, \infty) = \emptyset$, where $K$ is a compact subset of the real line. Moreover, the closure of the convex-polynomials on $[-1,b]$ are shown to be the functions that have a convex-power series representation. A continuous linear operator $T$ on a locally convex spa…
This paper presents Stieltjes-type integration for operator-valued functions with respect to spectral families. The relation between Riemann-Stieltjes integrals associated with some classes of spectral families including, in particular, those that arise in the context of the Stone representation theorem for Banach spaces is established. The developed approach is applied to the structural analysis of the operators commuting with a one-parameter representation satisfying the Generalized Stone theorem.
According to the fundamental work of Yu.V. Prokhorov, the general theory of stochastic processes can be regarded as the theory of probability measures in complete separable metric spaces. Since stochastic processes depending upon a continuous parameter are basically probability measures on certain subspaces of the space of all functions of a real variable, a particularly important case of this theory is when the underlying metric space has a linear structure. Prokhorov also provided a concrete metrisation of the topology of weak convergence today known as the L{é}vy-Prokhorov distance. Motivat…
Given $\u$ a multiplicative sequence of polynomial ideals, we consider the associated algebra of holomorphic functions of bounded type, $H_{b\u}(E)$. We prove that, under very natural conditions verified by many usual classes of polynomials, the spectrum $M_{b\u}(E)$ of this algebra "behaves" like the classical case of $M_{b}(E)$ (the spectrum of $H_b(E)$, the algebra of bounded type holomorphic functions). More precisely, we prove that $M_{b\u}(E)$ can be endowed with a structure of Riemann domain over $E"$ and that the extension of each $f\in H_{b\u}(E)$ to the spectrum is an $\u$-holomorphi…
In this note, we show that the spectral theorem, has two representations; the Stone-von Neumann representation and one based on the polar decomposition of linear operators, which we call the deformed representation. The deformed representation has the advantage that it provides an easy extension to all closed densely defined linear operators on Hilbert space. Furthermore, the deformed representation can also be extended all separable reflexive Banach spaces and has a limited extension to non-reflexive Banach spaces.
A compact space is said to be weakly Radon-Nikodým (WRN) if it can be weak*-embedded into the dual of a Banach space not containing $\ell_1$. We investigate WRN Boolean algebras, i.e. algebras whose Stone space is WRN compact. We show that the class of WRN algebras and the class of minimally generated algebras are incomparable. In particular, we construct a minimally generated nonWRN Boolean algebra whose Stone space is a separable Rosenthal compactum, answering in this way a question of W. Marciszewski. We also study questions of J. Rodríguez and R. Haydon concerning measures and the existenc…
We consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the spectra of weighted algebras and endow them with an analytic structure. We also deal with composition operators and algebra homomorphisms, in particular to investigate how their induced mappings act on the analytic structure of the spectrum. Moreover, a Banach-Stone type question is addressed.
It is well known that every (real or complex) normed linear space $L$ is isometrically embeddable into $C(X)$ for some compact Hausdorff space $X$. Here $X$ is the closed unit ball of $L^*$ (the set of all continuous scalar-valued linear mappings on $L$) endowed with the weak$^*$ topology, which is compact by the Banach-Alaoglu theorem. We prove that the compact Hausdorff space $X$ can indeed be chosen to be the Stone-Cech compactification of $L^*\setminus\{0\}$, where $L^*\setminus\{0\}$ is endowed with the supremum norm topology.
We describe a subclass of the class of normal operators on Banach spaces over non-Archimedean fields (A. N. Kochubei, J. Math. Phys. 51 (2010), article 023526) consisting of operators whose properties resemble those of unitary operators. In particular, an analog of Stone's theorem about one-parameter groups of unitary operators is proved.
Generalized almost disjoint families and injective Banach spaces
研究巴拿赫空间$c_0$的内射维数问题，通过引入拓扑空间上的广义几乎不交族概念，证明在连续统假设下$c_0$的内射维数至少为3。
Obstructed subhomogeneous-bundle extensions and embeddings
研究局部平凡次次齐 bundle 的全局扩展、次次齐 bundle 的同胚嵌入及局部平凡紧致李群等变 Banach 或 C* bundle 的恢复问题，提出在同调约束下扩展、嵌入及特征化的结果。
We study the extension of holomorphic functions of bounded type defined on an open subset of a Banach space, to larger domains. For this, we first characterize the envelope of holomorphy of a Riemann domain over a Banach space, with respect to the algebra of bounded type holomorphic functions, in terms of the spectrum of the algebra. We then give a simple description of the envelopes of balanced open sets and relate the concepts of domain of holomorphy and polynomial convexity. We show that for bounded balanced sets, extensions to the envelope are always of bounded type, and that this does not…
Isometries of James-Schreier and Lorentz spaces
本文通过极值点保持方法研究实 Banach 序列空间上的满射线性等距群，刻画了 James-Schreier 空间 $V_1$ 对偶单位球的极值点，证明其等距映射仅为 $\pm Id$，并给出 Lorentz 序列空间的 Banach-Stone 型结果。
Let $T:Y\to X$ be a bounded linear operator between two normed spaces. We characterize compactness of $T$ in terms of differentiability of the Lipschitz functions defined on $X$ with values in another normed space $Z$. Furthermore, using a similar technique we can also characterize finite rank operators in terms of differentiability of a wider class of functions but still with Lipschitz flavour. As an application we obtain a Banach-Stone-like theorem. On the other hand, we give an extension of a result of Bourgain and Diestel related to limited operators and cosingularity.
Let $(Ω,μ)$ be a $σ$-finite measure space. Given a Banach space $X$, let the symbol $S(X)$ stand for the unit sphere of $X$. We prove that the space $L^{\infty} (Ω,μ)$ of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm, satisfies the Mazur-Ulam property, that is, if $X$ is any complex Banach space, every surjective isometry $Δ: S(L^{\infty} (Ω,μ))\to S(X)$ admits an extension to a surjective real linear isometry $T: L^{\infty} (Ω,μ)\to X$. This conclusion is derived from a more general statement which assures that every surjective isometry …
We generalise Wigner's theorem to its most general form possible for B(h) in the sense of completely characterising those vector state transformations of B(h) that appear as restrictions of duals of linear operators on B(h). We then use this result to similarly characterise the pure state transformations of general C*-algebras that appear as restrictions of duals of linear operators on the underlying algebras. This result may be interpreted as a noncommutative Banach-Stone theorem.
We study the Banach algebras ${\rm C}(X, R)$ of continuous functions from a compact Hausdorff topological space $X$ to a Banach ring $R$ whose topology is discrete. We prove that the Berkovich spectrum of ${\rm C}(X, R)$ is homeomorphic to $ζ(X) \times {\mathcal M}(R)$, where $ζ(X)$ is the Banaschewski compactification of $X$ and ${\mathcal M}(R)$ is the Berkovich spectrum of $R$. We study how the topology of the spectrum of ${\rm C}(X, R)$ is related to the notion of homotopy Zariski open embedding used in derived geometry. We find that the topology of $ζ(X)$ can be easily reconstructed from …
Analogues of the classical Banach-Stone theorem for spaces of continuous functions are studied in the context of the spaces of absolutely continuous functions introduced by Ashton and Doust. We show that if $AC(σ_1)$ is algebra isomorphic to $AC(σ_2)$ then $σ_1$ is homeomorphic to $σ_2$. The converse however is false. In a positive direction we show that the converse implication does hold if the sets $σ_1$ and $σ_2$ are confined to a restricted collection of compact sets, such as the set of all simple polygons.
We say that a map $T: S_X\rightarrow S_Y$ between the unit spheres of two real normed-spaces $X$ and $Y$ is a phase-isometry if it satisfies \begin{eqnarray*} \{\|T(x)+T(y)\|, \|T(x)-T(y)\|\}=\{\|x+y\|, \|x-y\|\} \end{eqnarray*} for all $x,y\in S_X$. In the present paper, we show that there is a phase function $\varepsilon:S_X\rightarrow \{-1,1\}$ such that $\varepsilon \cdot T$ is an isometry which can be extended a linear isometry on the whole space $X$ whenever $T$ is surjective, $X=C(K)$ ($K$ is a compact Hausdorff space) and $Y$ is an arbitrary Banach space. Additionally, if $T$ is a phas…
On the relation between distances and seminorms on Fréchet spaces, with application to isometries
研究Fréchet空间上由半范数序列定义的度量，给出保持度量的线性算子保持各半范数的充分条件，并推广Banach-Stone定理至全纯与连续函数空间。
Let $X$ be a completely regular space. For a non-vanishing self-adjoint Banach subalgebra $H$ of $C_B(X)$ which has local units we construct the spectrum $\mathfrak{sp}(H)$ of $H$ as an open subspace of the Stone-Cech compactification of $X$ which contains $X$ as a dense subspace. The construction of $\mathfrak{sp}(H)$ is simple. This enables us to study certain properties of $\mathfrak{sp}(H)$, among them are various compactness and connectedness properties. In particular, we find necessary and sufficient conditions in terms of either $H$ or $X$ under which $\mathfrak{sp}(H)$ is connected, lo…
Operator Algebras of Bourgain Delbaen Spaces: Realization, Rigidity, and Ideal Structure
本文研究了Calkin代数及其作为紧算子商的结构，通过保持几何和拓扑信息实现交换C*代数。证明了对于每个紧度量空间K，存在反射Banach空间X_C(K)，其Calkin代数同构于C(K)。同时建立了稳定性、局部化原理、Hölder连续性、刚性定理及理想结构分类等贡献。
The classical theorems of Banach and Stone, Gelfand and Kolmogorov, and Kaplansky show that a compact Hausdorff space $X$ is uniquely determined by the linear isometric structure, the algebraic structure, and the lattice structure, respectively, of the space $C(X)$. In this paper, it is shown that for rather general subspaces $A(X)$ and $A(Y)$ of $C(X)$ and $C(Y)$ respectively, any linear bijection $T: A(X) \to A(Y)$ such that $f \geq 0$ if and only if $Tf \geq 0$ gives rise to a homeomorphism $h: X \to Y$ with which $T$ can be represented as a weighted composition operator. The three classica…
Structure properties of Banach spaces of I-null sequences
本文研究理想零序列空间c_{0,I}及其相关商空间的结构性质，通过Dini原理和Sobczyk定理等工具，刻画了对偶空间、紧性、补余拷贝等经典问题，并构造了统计理想以展示非补余现象。
It is known that the classical Banach--Stone theorem does not extend to the class of $AC(σ)$ spaces of absolutely continuous functions defined on compact subsets of the complex plane. On the other hand, if $σ$ is restricted to the set of compact polygons, then all the corresponding $AC(σ)$ spaces are isomorphic. In this paper we examine the case where $σ$ is the spectrum of a compact operator, and show that in this case one can obtain an infinite family of homeomorphic sets for which the corresponding function spaces are not isomorphic.
Let $Ω$ be a smooth compact oriented 3-dimensional Riemannian manifold with boundary. A quaternion field is a pair $q=\{α,u\}$ of a function $α$ and a vector field $u$ on $Ω$. A field $q$ is {\it harmonic} if $α, u$ are continuous in $Ω$ and $\nablaα={\rm rot\,}u,\,{\rm div\,}u=0$ holds into $Ω$. The space ${\mathscr Q}(Ω)$ of harmonic fields is a subspace of the Banach algebra $\mathscr C\left(Ω\right)$ of continuous quaternion fields with the point-wise multiplication $qq'=\{αα'-u\cdot u',\,αu'+α'u+u\wedge u'\}$. We prove a Stone-Weierstrass type theorem: the subalgebra $\vee{\mathscr Q}(Ω)$…
Let $X$ be a (topological) space and let ${\mathscr I}$ be an ideal in $X$, that is, a collection of subsets of $X$ which contains all subsets of its elements and is closed under finite unions. The elements of ${\mathscr I}$ are called null. The space $X$ is locally null if each $x$ in $X$ has a null neighborhood in $X$. Let $C_b(X)$ denote the normed algebra of all continuous bounded real-valued mappings on $X$ equipped with the supremum norm, $C_0(X)$ denote the subalgebra of $C_b(X)$ consisting of elements vanishing at infinity and $C_{00}(X)$ the subalgebra of $C_b(X)$ consisting of elemen…
Let C(K) denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space K. For two subsets of C(K), one can define their product by pointwise multiplication, just as the Minkowski sum of the sets is defined by pointwise addition. Our main interest is in correlations between properties of the product of closed order intervals in C(K) and properties of the underlying space K. When K is finite, the product of two intervals in C(K) is always an interval. Surprisingly, the converse of this result is true for a wide class of compacta. We show that a firs…
We consider operators acting on a UMD Banach lattice $X$ that have the same algebraic structure as the position and momentum operators associated with the harmonic oscillator $-\frac12Δ+ \frac12|x|^{2} $ acting on $L^{2}(\mathbb{R}^{d})$. More precisely, we consider abstract harmonic oscillators of the form $\frac12 \sum _{j=1} ^{d}(A_{j}^{2}+B_{j}^{2})$ for tuples of operators $A=(A_{j})_{j=1} ^{d}$ and $B=(B_{k})_{k=1} ^{d}$, where $iA_j$ and $iB_k$ are assumed to generate $C_{0}$ groups and to satisfy the canonical commutator relations. We prove functional calculus results for these abstrac…
Isometries between spaces of metrics
本文证明度量空间上的Banach-Stone型定理：可度量空间X和Y同胚当且仅当它们的有界连续伪度量空间或可容许度量空间之间存在保持特定子空间的满射等距，且等距由同胚诱导。
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