We construct a compact linearly ordered space $K$ of weight aleph one, such that the space $C(K)$ is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, $K$ is a continuous image of a Valdivia compact and every separable subspace of $C(K)$ is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.
A projectional skeleton in a Banach space is a sigma-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Sigma-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of kno…
We show a way of constructing projectional skeletons using the concept of rich families in Banach spaces which admit a projectional generator. Our next result is that a Banach space $X$ is Asplund and weakly compactly generated if and only if there exists a commutative 1-projectional skeleton $(Q_γ:\ γ\inΓ)$ on $X$ such that $(Q_γ^*:\ γ\inΓ)$ is a commutative 1-projectional skeleton on $X^*$. We consider both, real and also complex, Banach spaces.
We list a number of problems in several topics related to compactness in nonseparable Banach spaces. Namely, about the Hilbertian ball in its weak topology, spaces of continuous functions on Eberlein compacta, WCG Banach spaces, Valdivia compacta and Radon-Nikodým compacta.
We study projectional skeletons and the Plichko property in Lipschitz-free spaces, relating these concepts to the geometry of the underlying metric space. Specifically, we identify a metric property that characterizes the Plichko property witnessed by Dirac measures in the associated Lipschitz-free space. We also show that the Lipschitz-free space of all $\mathbb{R}$-trees has the Plichko property witnessed by molecules, and define the concept of retractional trees to generalize this result to a bigger class of metric spaces. Finally, we show that no separable subspace of $\ell_\infty$ contain…
We prove that if X is an infinite-dimensional Banach space with C^p smooth partitions of unity, then X and X\K are C^p diffeomorphic, for every weakly compact subset K of X.
Pointwise function spaces over compacta are not weak Banach spaces
Czech Academy of Sciences（捷克科学院）；Institute of Mathematics and Computer Science Jagiellonian University（雅盖隆大学数学与计算机科学研究所）；Faculty of Mathematics and Computer Science Adam Mickiewicz University（亚当·密茨凯维奇大学数学与计算机科学系）
We prove that Banach spaces with a $1$-projectional skeleton form a $\mathcal{P}$-class and deduce that any such space admits a strong Markushevich basis. We provide several equivalent characterizations of spaces with a projectional skeleton and of spaces having a commutative one. We further analyze known examples of spaces with a non-commutative projectional skeleton and compare their behavior with the commutative case. Finally, we collect several open problems.
We provide an alternative proof of the theorem saying that any Vašák (or, weakly countably determined) Banach space admits a full $1$-projectional skeleton. The proof is done with the use of the method of elementary submodels and is comparably simple as the proof given by W.~Kubiś (2009) in case of weakly compactly generated spaces.
We show that for every Dugundji compact $K$ of weight aleph one the Banach space $C(K)$ is 1-Plichko and the space $P(K)$ of probability measures on $K$ is Valdivia compact. Combining this result with the existence of a non-Valdivia compact group, we answer a question of Kalenda.
We find characterizations of Vašák spaces and weakly $\mathcal{K}$-analytic spaces using the notions of separable projectional resolution of the identity (SPRI) and of projectional skeleton. This in particular addresses a recent challenge suggested by M. Fabian and V. Montesinos in \cite{FM18}. Our method of proof also gives similar characterizations of WCG spaces and their subspaces (some aspects of which were known, some are new). Moreover we show that for countably many projectional skeletons $\{\mathfrak{s}_n: n \in ω\}$ on a Banach space inducing the same set, there exists a projectional …
We show that there are no automorphic Banach spaces of the form C(K) with K continuous image of Valdivia compact except the spaces c0(I). Nevertheless, when K is an Eberlein compact of finite height such that C(K) is not isomorphic to c0(I), all isomorphism between subspaces of C(K) of size less than aleph_omega extend to automorphisms of C(K).
A Banach space $E$ is said to be injective if for every Banach space $X$ and every subspace $Y$ of $X$ every operator $t:Y\to E$ has an extension $T:X\to E$. We say that $E$ is $\aleph$-injective (respectively, universally $\aleph$-injective) if the preceding condition holds for Banach spaces $X$ (respectively $Y$) with density less than a given uncountable cardinal $\aleph$. We perform a study of $\aleph$-injective and universally $\aleph$-injective Banach spaces which extends the basic case where $\aleph=\aleph_1$ is the first uncountable cardinal. When dealing with the corresponding "isomet…
We prove that if K is a Gruenhage compact space then C(K)* admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and X* is the |.|-closed linear span of K, where K is a Gruenhage compact in the w*-topology and |.| is equivalent to a coarser, w*-lower semicontinuous norm on X*, then X* admits an equivalent, strictly convex dual norm. We give a partial converse to the first result by showing that if T is a tree, then C(T)* admits an equivalent, strictly convex dual norm if and only if T is a Gruenhage space. Finally, we present some stability proper…
A compact space is said to be weakly Radon-Nikodým if it is homeomorphic to a weak*-compact subset of the dual of a Banach space not containing an isomorphic copy of $\ell_1$. In this work we provide an example of a continuous image of a Radon-Nikodým compact space which is not weakly Radon-Nikodým. Moreover, we define a superclass of the continuous images of weakly Radon-Nikodým compact spaces and study its relation with Corson compacta and weakly Radon-Nikodým compacta.
We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta $K_{\mathcal A}$ induced by almost disjoint families ${\mathcal A}$ of countable subsets of uncountable sets. For these spaces, we prove among others that $C(K_{\mathcal A})$ has the controlled variant of the separable complementation property if and only if $C(K_{\mathcal A})$ is Lindelöf in the weak topology if and only if $K_{\mathcal A}$ is monolithic. We give an example of ${\mathcal A}$ for which $C(K_{\mathcal A})$ has the SCP, while $K_{\mathcal A}$ is…
We prove some generalizations of results concerning Valdivia compact spaces (equivalently spaces with a commutative retractional skeleton) to the spaces with a retractional skeleton (not necessarily commutative). Namely, we show that the dual unit ball of a Banach space is Corson provided the dual unit ball of every equivalent norm has a retractional skeleton. Another result to be mentioned is the following. Having a compact space K, we show that K is Corson if and only if every continuous image of K has a retractional skeleton.
We study the class of compact spaces that appear as structure spaces of separable Banach lattices. In other words, we analyze what $C(K)$ spaces appear as principal ideals of separable Banach lattices. Among other things, it is shown that every such compactum $K$ admits a strictly positive regular Borel measure of countable type that is analytic, and in the nonmetrizable case these compacta are saturated with copies of $β\mathbb N$. Some natural questions about this class are left open.
We investigate the problem of classifying the Banach spaces $\mathrm{Lip}_0(C(K))$ for Hausdorff compacta $K$. In particular, sufficient conditions are established for a space $\mathrm{Lip}_0(C(K))$ to be isomorphic to $\mathrm{Lip}_0(c_0(\varGamma))$ for some uncountable set $\varGamma$.
Two applications of Nash-Williams' theory of barriers to sequences on Banach spaces are presented: The first one is the $c_0$-saturation of $C(K)$, $K$ countable compacta. The second one is the construction of weakly-null sequences generalizing the example of Maurey-Rosenthal.
In the present paper we introduce and study the Lipschitz retractional structure of metric spaces. This topic was motivated by the analogous projectional structure of Banach spaces, a topic that has been thoroughly investigated. The more general metric setting fits well with the currently active theory of Lipschitz free spaces and spaces of Lipschitz functions. Among our applications we show that the Lipschitz free space $\mathcal{F}(X)$ is a Plichko space whenever $X$ is a Plichko Banach space. Our main results include two examples of metric spaces. The first one $M$ contains two points $\{0,…
It is shown that every normalized weakly null sequence of length $κ_λ$ in a Banach space has a subsequence of length $λ$ which is an unconditional basic sequence; here $κ_λ$ is a large cardinal depending on a given infinite cardinal $λ$. Transfinite topological games on Banach spaces are analyzed which determine the existence of a long unconditional basic sequence. Then 'asymptotic disentanglement' condition in a transfinite setting is studied which ensures a winning strategy for the unconditional basic sequence builder in the above game. The following problem is investigated: When does a Mark…
We present selected known results and some of their improvements, involving Gurarii spaces. A Banach space is Gurarii if it has certain natural extension property for almost isometric embeddings of finite-dimensional spaces. Deleting the word "almost", we get the notion of a strong Gurarii space. There exists a unique (up to isometry) separable Gurarii space, however strong Gurarii spaces cannot be separable. The structure of the class of non-separable Gurarii spaces seems to be not very well understood. We discuss some of their properties and state some open questions. In particular, we chara…
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 investigate the geometry of $C(K,X)$ and $\ell_{\infty}(X)$ spaces through complemented subspaces of the form $\left(\bigoplus_{i\in \varGamma}X_i\right)_{c_0}$. Concerning the geometry of $C(K,X)$ spaces we extend some results of D. Alspach and E. M. Galego from \cite{AlspachGalego}. On $\ell_{\infty}$-sums of Banach spaces we prove that if $\ell_{\infty}(X)$ has a complemented subspace isomorphic to $c_0(Y)$, then, for some $n \in \mathbb{N}$, $X^n$ has a subspace isomorphic to $c_0(Y)$. We further prove the following: (1) If $C(K)\sim c_0(C(K))$ and $C(L)\sim c_0(C(L))$ and $\ell_{\infty…
We investigate the following problem posed by Cabello Sanchéz, Castillo, Kalton, and Yost: Let $K$ be a nonmetrizable compact space. Does there exist a nontrivial twisted sum of $c_0$ and $C(K)$, i.e., does there exist a Banach space $X$ containing a non-complemented copy $Z$ of $c_0$ such that the quotient space $X/Z$ is isomorphic to $C(K)$? Using additional set-theoretic assumptions we give the first examples of compact spaces $K$ providing a negative answer to this question. We show that under Martin's axiom and the negation of the continuum hypothesis, if either $K$ is the Cantor cube $2^…
