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}.
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.
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.
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 $φ$ …
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.
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…
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…
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 $C(Y)$.
Mathematical Institute, University of Wrocław（弗罗茨瓦夫大学数学研究所）；Faculty of Mathematics, Informatics and Mechanics, University of Warsaw（华沙大学数学、信息学和力学学院）；Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）
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.
We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties which asserted the existence of an uncountable independent family. In fact we prove it for a bigger class of Boolean algebras satisfying much weaker properties. It follows that the Stone spaces of all such Boolean algebras contains a copy of the Cech-Stone compactification of the integers and the Banach space of contnuous functions on them has $l_\infty$ as a quotient. Connections with the Grothendieck property in Ban…
We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.
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*).
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.
Suppose that $(x_s)_{s\in S}$ is a normalized family in a Banach space indexed by the dyadic tree $S$. Using Stern's combinatorial theorem we extend important results from sequences in Banach spaces to tree-families. More precisely, assuming that for any infinite chain $β$ of $S$ the sequence $(x_s)_{s\inβ}$ is weakly null, we prove that there exists a subtree $T$ of $S$ such that for any infinite chain $β$ of $T$ the sequence $(x_s)_{s\inβ}$ is nearly (resp., convexly) unconditional. In the case where $(f_s)_{s\in S}$ is a family of continuous functions, under some additional assumptions, we …
Generalized almost disjoint families and injective Banach spaces
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）；Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）
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…
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…
A subset $A$ of a Banach space is called Banach-Saks when every sequence in $A$ has a Ces{à}ro convergent subsequence. Our interest here focusses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By means of a combinatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result.
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…
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 survey, we present several results related to characterizing the surjective isometries on Banach sequence spaces. Our survey includes full proofs of these characterizations for the classical spaces as well as more recent results for combinatorial Banach spaces and Tsirelson-type spaces. Along the way, we pose many open problems related to the structure of the group of surjective isometries and characterizations of the group of surjective isometries for various Banach spaces.
This paper is concerned with the isomorphic structure of the Banach space $\ell_\infty/c_0$ and how it depends on combinatorial tools whose existence is consistent but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that $\ell_\infty/c_0$ does not have an orthogonal \break $\ell_\infty$-decomposition that is, it is not of the form $\ell_\infty(X)$ for any Banach space $X$. The main local result is that it is consistent that $\ell_\infty(c_0(\mathfrak{c}))$ does not embed isomorphically into $\ell_\infty/c_0$, where $\mathfrak{c}$ is the cardinality of…
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…
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 …
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.
We prove that every Banach space containing a subspace isomorphic to $\co$ fails the fixed point property. The proof is based on an amalgamation approach involving a suitable combination of known results and techniques, including James's distortion theorem, Ramsey's combinatorial theorem, Brunel-Sucheston spreading model techniques and Dowling, Lennard and Turett's fixed point methodology employed in their characterization of weak compactness in $\co$.
We show that every Banach space $X$ containing an isomorphic copy of $c_0$ has an infinite equilateral set and also that if $X$ has a bounded biorthogonal system of size $α$ then it can be renormed so as to admit an equilateral set of equal size. If $K$ is any compact non metrizable space, then under a certain combinatorial condition on $K$ the Banach space $C(K)$ has an uncountable equilateral set.
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.
We present a construction that allows us to define a limit object of Banach space decorated graph sequences in a generalized homomorphism density sense. This general functional analytic framework provides a universal language for various combinatorial limit notions. In particular it makes it possible to assign limit objects to multigraph sequences that are convergent in the sense of node-and-edge homomorphism numbers, and it generalizes the limit theory for graph sequences with compact decorations.
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 …
The paper is concerned with the problem whether a nonseparable Banach space must contain an uncountable set of vectors such that the distances between every two distinct vectors of the set are the same. Such sets are called equilateral. We show that Martin's axiom and the negation of the continuum hypothesis imply that every nonseparable Banach space of the form $C(K)$ has an uncountable equilateral set. We also show that one cannot obtain such a result without an additional set-theoretic assumption since we construct an example of nonseparable Banach space of the form $C(K)$ which has no unco…
We study delta-points in Banach spaces $h_{\mathcal{A},p}$ generated by adequate families $\mathcal A$ where $1 \le p < \infty$. In the case the familiy $\mathcal A$ is regular and $p=1,$ these spaces are known as combinatorial Banach spaces. When $p > 1$ we prove that neither $h_{\mathcal{A},p}$ nor its dual contain delta-points. Under the extra assumption that $\mathcal A$ is regular, we prove that the same is true when $p=1.$ In particular the Schreier spaces and their duals fail to have delta-points. If $\mathcal A$ consists of finite sets only we are able to rule out the existence of delt…
Szemerédi's regularity lemma is a fundamental tool in extremal graph theory, theoretical computer science and combinatorial number theory. Lovász and Szegedy [L. Lovász and B. Szegedy: Szemerédi's Lemma for the analyst, Geometric and Functional Analysis 17 (2007), 252-270] gave a Hilbert space interpretation of the lemma and an interpretation in terms of compact- ness of the space of graph limits. In this paper we prove several compactness results in a Banach space setting, generalising results of Lovász and Szegedy as well as a result of Borgs, Chayes, Cohn and Zhao [C. Borgs, J.T. Chayes, H.…
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 continue our investigation, from \cite{dh}, of the ring-theoretic infiniteness properties of ultrapowers of Banach algebras, studying in this paper the notion of being purely infinite. It is well known that a $C^*$-algebra is purely infinite if and only if any of its ultrapowers are. We find examples of Banach algebras, as algebras of operators on Banach spaces, which do have purely infinite ultrapowers. Our main contribution is the construction of a "Cuntz-like" Banach $*$-algebra which is purely infinite, but whose ultrapowers are not even simple, and hence not purely infinite. This algeb…
We show that the Gurarij space $\mathbb{G}$ has extremely amenable automorphism group. This answers a question of Melleray and Tsankov. We also compute the universal minimal flow of the automorphism group of the Poulsen simplex $\mathbb{P}$ and we prove that it consists of the canonical action on $\mathbb{P}$ itself. This answers a question of Conley and Törnquist. We show that the pointwise stabilizer of any closed proper face of $\mathbb{P}$ is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simp…
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.
Structure properties of Banach spaces of I-null sequences
Escuela de Matemáticas, Universidad Industrial de Santander（桑坦德工业大学生命科学学院）；Department of Biotechnology and Bioprocess, São Paulo State University（圣保罗州立大学生物技术与生物工艺系）；Universidad Industrial de Santander（桑坦德工业大学）
