Related papers: $\aleph$-injective Banach spaces and $\aleph$-proj…

In the paper compact multiplier operators on Banach spaces of analytic functions on the unit disk with the range in Banach sequence lattices are studied. If the domain space $X$ is such that $H_\infty\hookrightarrow X\hookrightarrow H_1$,…

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…

In a previous paper the second author introduced a compact topology on the space of closed ideals of a unital Banach algebra A. If A is separable then this topology is either metrizable or else neither Hausdorff nor first countable. Here it…

We prove that for any $\ell_\infty$-sum $Z = \bigoplus_{i \in [n]} X_i$ of finitely many strictly convex Banach spaces $(X_i)_{i \in [n]}$, an extremeness preserving 1-Lipschitz bijection $f\colon B_Z \to B_Z$ is an isometry, by…

We study biorthogonal sequences with special properties, such as weak or weak-star convergence to 0, and obtain an extension of the Josefson-Nissenzweig theorem. This result is applied to embed analytic disks in the fiber over 0 of the…

Recent developments in Banach space theory provided unexpected examples of unital Banach algebras that are isomorphic to Calkin algebras of Banach spaces, however no example of a unital Banach algebra that cannot be realised as a~Calkin…

We show that for "most" compact non metrizable spaces, the unit ball of the Banach space C(K) contains an uncountable 2-equilateral set. We also give examples of compact non metrizable spaces K such that the minimum cardinality of a maximal…

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…

For every $\alpha<\omega_1$ we establish the existence of a separable Banach space whose Szlenk index is $\omega^{\alpha\omega+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed…

We characterize those classes $\ccc$ of separable Banach spaces admitting a separable universal space $Y$ (that is, a space $Y$ containing, up to isomorphism, all members of $\ccc$) which is not universal for all separable Banach spaces.…

Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z) \ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any…

We study Banach spaces $C(K)$ of real-valued continuous functions from the finite product of compact lines. It turns out that the topological character of these compact lines can be used to distinguish whether two spaces of continuous…

Let $E$ be a Banach space, and $\mathcal B(E)$ the algebra of all bounded linear operators on $E$. The question of amenability of $\mathcal B(E)$ goes back to Johnson's seminal memoir \cite{johnson} from 1972. We present the first general…

Classes of Banach spaces that are finitely, strongly finitely or elementary equivalent are introduced. On sets of these classes topologies are defined in such a way that sets of defined classes become compact totally disconnected…

In this paper, we investigate the notion $\phi$-injectivity for Banach $A$-modules, where $\phi$ is a character on $A.$ We obtain some hereditary properties of $\phi$-injectivity for certain classes of Banach modules related to closed…

We initiate a study of structural properties of the quotient algebra $\mathcal K(X)/\mathcal A(X)$ of the compact-by-approximable operators on Banach spaces $X$ failing the approximation property. Our main results and examples include the…

For a Banach space $X$ its subset $Y\subseteq X$ is called overcomplete if $|Y|=dens(X)$ and $Z$ is linearly dense in $X$ for every $Z\subseteq Y$ with $|Z|=|Y|$. In the context of nonseparable Banach spaces this notion was introduced…

In this paper we establish some new results concerning the Cauchy-Peano problem in Banach spaces. Firstly, we prove that if a Banach space $E$ admits a fundamental biorthogonal system, then there exists a continuous vector field $f\colon…

We show that a QWEP von Neumann algebra has the weak* positive approximation property if and only if it is seemingly injective in the following sense: there is a factorization of the identity of $M$ $$Id_M=vu: M{\buildrel…

We prove a number of results concerning the embedding of a Banach lattice $X$ into an r.i. space $Y$. For example we show that if $Y$ is an r.i. space on $[0,\infty)$ which is $p$-convex for some $p>2$ and has nontrivial concavity then any…