Related papers: On Unconditionality and Higher-Order Schreier Unco…

Let $\A$ be the operator which assigns to each $m \times n$ matrix-valued function on the unit circle with entries in $H^\infty + C$ its unique superoptimal approximant in the space of bounded analytic $m \times n$ matrix-valued functions…

It is consistent that the continuum be arbitrary large and no absolute $\kappa$-Borel set $X$ of density $\kappa$, $\aleph_1<\kappa<\mathfrak{c}$, condenses onto a compact metric space. It is consistent that the continuum be arbitrary large…

A base of a topological space is called {\em Noetherian } iff it does not contain an infinite strictly $\subseteq$-increasing chain. We show that minimal cardinality of a regular spaces without a Noetherian base is the first strongly…

We present a reflexive Banach space with an unconditional basis which is quasi-minimal and tight by range, i.e. of type (4) in Ferenczi-Rosendal list within the framework of Gowers' classification program of Banach spaces, but contrary to…

The ``multiple of the inclusion plus compact problem'' which was posed by T.W. Gowers in 1996 and Th. Schlumprecht in 2003, asks whether for every infinite dimensional Banach space $X$ there exists a closed subspace $Y$ of $X$ and a bounded…

The Erberlein-Smulian Theorem asserts that for complete normed spaces, that is Banach spaces, a subset is weak compact if and only if it is weak sequentially compact. In this paper it is shown that the completeness of the normed space is…

Under large cardinal hypotheses beyond the Kunen inconsistency -- hypotheses so strong as to contradict the Axiom of Choice -- we solve several variants of the generalized continuum problem and identify structural features of the levels…

We consider first-order logic over the subword ordering on finite words, where each word is available as a constant. Our first result is that the $\Sigma_1$ theory is undecidable (already over two letters). We investigate the decidability…

We prove that for any integers $\alpha, \beta > 1$, the existential fragment of the first-order theory of the structure $\langle \mathbb{Z}; 0,1,<, +, \alpha^{\mathbb{N}}, \beta^{\mathbb{N}}\rangle$ is decidable (where $\alpha^{\mathbb{N}}$…

Let $A$ be a (not necessarily unital) separable non-elementary simple amenable C*-algebra whose tracial basis may not have finite covering dimension and may not be compact but satisfies certain condition (C). We show that $A$ is ${\cal…

We prove Los conjecture = Morley theorem in ZF, with the same characterization (of first order countable theories categorical in aleph_alpha for some (equivalently for every) ordinal alpha>0. Another central result here is, in this context:…

We show that a bounded quasinilpotent operator $T$ acting on an infinite dimensional Banach space has an invariant subspace if and only if there exists a rank one operator $F$ and a scalar $\alpha\in\mathbb{C}$, $\alpha\neq 0$, $\alpha\neq…

We consider a family of conditional nonlinear expectations defined on the space of bounded random variables and indexed by the class of all the sub-sigma-algebras of a given underlying sigma-algebra. We show that if this family satisfies a…

We extend the methods used by V. Ferenczi and Ch. Rosendal to obtain the `third dichotomy' in the program of classification of Banach spaces up to subspaces, in order to prove that a Banach space E with an admissible system of blocks with…

We show that a pseudoconvex open subset of a Banach space with an unconditional basis is biholomorphic to a closed direct submanifold of a Banach space with an unconditional basis.

Suppose that $X$ is a Banach space. We will show that $X$ does not contain a copy of $c_0$ if and only if for each series which is not unconditionally convergent in $X$ respective sets coding all bounded subseries and rearrangements are…

We define a topological ring $R$ to be \emph{Hirsch}, if for any unconditionally convergent series $\sum_{n\in\omega} x_i$ in $R$ and any neighborhood $U$ of the additive identity $0$ of $R$ there exists a neighborhood $V\subseteq R$ of $0$…

We consider generators of positive $C_0$-semigroups and, more generally, resolvent positive operators $A$ on ordered Banach spaces and seek for conditions ensuring the negativity of their spectral bound $s(A)$. Our main result characterizes…

We show that a plane continuum X is indecomposable iff X has a sequence (U_n) of not necessarily distinct complementary domains satisfying what we call the double-pass condition: If one draws an open arc A_n in each U_n whose ends limit…

We prove that if $ C $ is a family of separable Banach spaces which is analytic with respect to the Effros-Borel structure and none member of $ C $ is isometrically universal for all separable Banach spaces, then there exists a separable…