Hereditarily indecomposable Banach spaces may have density at most continuum (Plichko-Yost, Argyros-Tolias). In this paper we show that this cannot be proved for indecomposable Banach spaces. We provide the first example of an indecomposable Banach space of density two to continuum. The space exists consistently, is of the form C(K) and it has few operators in the sense that any bounded linear operator T on C(K) satisfies T(f)=gf+S(f) for every f in C(K), where g is in C(K) and S is weakly compact (strictly singular).

We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta K_A induced by almost disjoint families A of countable subsets of uncountable sets. For these spaces, we prove among others that C(K_A) has the controlled variant of the separable complementation property if and only if C(K_A) is Lindelof in the weak topology if and only if K_A is monolithic. We give an example of A for which C(K_A) has the SCP, while K_A is not monolithic and an example of a space C(K_A) with controlled and continuous SCP which has neither a projectional skeleton nor a projectional resolution of the identity. Finally, we describe the structure of almost disjoint families of cardinality omega-one which induce monolithic spaces of the form K_A: They can be obtained from countably many ladder systems and pairwise disjoint families applying simple operations.

For k being the first uncountable cardinal w₁ or k being the cardinality of the continuum c, we prove that it is consistent that there is no Banach space of density k in which it is possible to isomorphically embed every Banach space of the same density which has a uniformly Gateaux differentiable renorming or, equivalently, whose dual unit ball with the weak^* topology is a subspace of a Hilbert space (a uniform Eberlein compact space). This complements a consequence of results of M. Bell and of M. Fabian, G. Godefroy, V. Zizler that assuming the continuum hypothesis, there is a universal space for all Banach spaces of density k=c=w₁ which have a uniformly Gateaux differentiable renorming. Our result implies, in particular, that betaN-N may not map continuously onto a compact subset of a Hilbert space with the weak topology of density k=w₁ or k=c and that a C(K) space for some uniform Eberlein compact space K may not embed isomorphically into l_infty/c_₀.

We construct an infinite dimensional Banach space of continuous functions C(K) such that every one-to-one operator on C(K) is onto.

Accepted to Bulletin of the London Mathematical Society
For preprint click on the icon of the journal

Denote by [0,w₁] the compact Hausdorff space consisting of all ordinals not exceeding the first uncountable ordinal w₁, equipped with the order topology. We show that the K-groups of the Banach algebra $B(C[0,w₁]) of bounded operators on the Banach space C[0,w₁] of complex-valued, continuous functions on [0,w₁] are given by K_0(B(C[0,w₁]))=Z and K_1(B(C[0,w₁])) = {0}.

Denote by K the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let C be the Banach space of scalar-valued, continuous functions which are defined on K and vanish eventually. We show that a weakly* compact subset of the dual space of C is either uniformly Eberlein compact, or it contains a homeomorphic copy of the ordinal interval [0,w₁] Using this result, we deduce that a Banach space which is a quotient of C can either be embedded in a Hilbert-generated Banach space, or it is isomorphic to the direct sum of C and a subspace of a Hilbert-generated Banach space. Moreover, we obtain a list of eight equivalent conditions describing the Loy--Willis ideal, which is the unique maximal ideal of the Banach algebra of bounded, linear operators on C. As a consequence, we find that this ideal has a bounded left approximate identity, thus solving a problem left open by Loy and Willis, and we give new proofs, in some cases of stronger versions, of several known results about the Banach space C and the operators acting on it.

Workshop on set theoretic methods in compact spaces and Banach spaces, Warsaw, April 17-21, 2013

This conference will focus on applying set-theoretic analysis in the interactions between Banach spaces and compact spaces. Every Banach space X induces the dual ball BX* which is compact in the weak* topology and through which X can be sucessfully investigated. On the other hand every compact space K induces the Banach space C(K) of continuous functions on K, the construction which has been used to produce numerous conterexamples in the general Banach space theory. Combinatorial analysis of the geometry of Banach spaces and set-theoretic topology in the context of weak* and weak topologies associated with Banach spaces are the methods which turned out to be particularly sucessful in investigating the above intereactions.

Página do congresso

C*-algebras and Banach algebras, Warsaw, July 8-12, 2013

The meeting is planned as a part of activities of the Quantum groups, operators and non-commutative probability research network involving the Lancaster University, University of Leeds and IMPAN,
The speakers will include Marek Bożejko* (University of Wrocław), H.Garth Dales (Lancaster University), Matthew Daws (Leeds University), Biswarup Das (Leeds University), Derek Kitson (Lancaster University), Niels Laustsen (Lancaster University), Piotr Nowak (IMPAN and University of Warsaw), Anna Pelczar-Barwacz (Jagiellonian University), Grzegorz Plebanek (University of Wrocław), Marcin Sabok (IMPAN), Piotr Sołtan (University of Warsaw), Stephen Trotter (Leeds University), Stanisław Lech Woronowicz (University of Warsaw) and Wiesław Żelazko (IMPAN).
We plan 2 days of lectures and three days of research activities in smaller groups.

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Quantum groups, operators and non-commutative probability research network, a collaboration between Lancaster, Leeds and IMPAN Warsaw

TEMAS DE PESQUISA

• Quantum groups
• Quantum harmonic analysis
• Banach algebras
• Banach spaces and operators
• Stochastic analysis and non-commutative probability
• Set-theoretic analysis

Página da rede

O âmago do seminário: Métodos conjuntistas em variedade de disciplinas de matemática em particular, em espaços de Banach, C*-álgebras, topologia, estrutura da reta real, teoria da medida, álgebras de Boole, grupos topológicos. Aqui nos contamos tanto o desenvolvimento de métodos como forcing, teoria dos conjuntos descritíva, teoria de Ramsey, quanto aplicaçoes concretas deles nas diciplinas mencionados acima.

O caráter do grupo de trabalho: Vamos tentar manter um caráter mais do trabalho em curso do que de uma apresentaçao final. Isso significa que promovemos uma discuçao, digreçao, participaçao ativa. Para isso a duraçao do seminário nao é definido rigidamente e pode alcançar até 3 horas com intervalos. Queriamos poder nos aprofundar nos detalhes dos argumentos matemáticos estudados. Esperamos que isso pode ser um bom completamento dos seminários da quarta-feira da Universidade de Varsóvia.

A página www do seminário

Página do programa de doutorado do Instituto Matemático da Academia Polonesa de Ciencias

1. Colaboraçao com Profa. Christina Brech
2. Palestra no IME USP: 20.02.2013; Famílias independentes em algumas álgebras de Boole e separaçoes em espaços de Banach
3. Palestra no IME USP: 25.02.2013; Um espaço de Banach onde todo operador injectivo é surjetivo
4. Palestra no IME UNICAMP: 05.03.2013; Propriedades da complementaçao em espaços de Banach induzidos por famílias quase disjuntas

The inaugural meeting of the QOP network will be held at Lancaster University. The local organiser is Professor Garth Dales. This meeting is supported by the Visitor Fund of the Department of Mathematics and Statistics, Lancaster University.

The webpage of the meeting

RESEARCH WORKSHOP OF THE ISRAEL SCIENCE FOUNDATION

Interactions between Logic, Topological structures and Banach spaces theory

May 19-24, 2013

Eilat Campus of Ben-Gurion University of the Negev The webpage of the conference

The Hungarian Academy of Sciences, the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, the Eötvös Loránd University and the János Bolyai Mathematical Society announce that a conference dedicated to the 100th anniversary of Paul Erdős will be held from Monday, July 1st to Friday, July 5th, 2013, in Budapest, Hungary. The distinguished main patron of the conference is József Pálinkás, the President of the Hungarian Academy of Sciences. The topic of the conference includes all basic fields that Paul Erdős contributed to: Number Theory, Combinatorics (including Combinatorial Algebra, Combinatorial Geometry and Theoretical Computer Science), Analysis (including Approximation Theory and Ergodic Theory), Probability Theory, and Set Theory among others. The main goal of the conference is to explore Paul Erdős' wide ranging contributions to mathematics, and to survey the trends of development originating in his work.

The webpage of the conference

The field experienced dramatic developments in recent years.
A new approach to consistency lower bounds in set theory, which one might refer to as quasi lower bounds, emerged initially from work of Neeman and has been developed further by Friedman-Holy, Sakai and Velickovic and especially Viale-Weiß. Instead of showing that large cardinals are definitively required for set-theoretic properties, one shows that they are required to obtain those properties by the method of forcing, given certain hypotheses on the ground model and the type of forcing used.
A second development is in the application of descriptive set theory to the study of C*-algebras: Farah showed that it is consistent that all automorphisms of the Calkin algebra are inner, and further joint work of his with Toms, Törnquist and others has recently produced dramatic results regarding the unclassifiability of separable C*-algebras. A striking interaction of set theory with ergodic theory is Foreman-Weiss's recent anti-classification theorem for measure-preserving diffeomorphisms of the torus. The ESI Programme will bring together well established research leaders of the field as well as young postdocs and PhD students to work on these (and other) developments.

The webpage of the conference

Membro da banca editorial Extracta Mathematicae

Áreas: Analytic and set theoretic topology, Banach spaces of continuous functions, combinatorial set theory

March 19 and 26 at 12.00, Room 105. Piotr Koszmider; Remarks on representations of the bidual to C(T).

Abstract: Continuing our attempts of understanding the space of multiplicative functionals on M(T) (the measure algebra on the circle with convolution) with the weak* topology we will present some classical results concerning the representation of the bidual of C(T) as some C(K). We will also discuss another representation due to Mauldin which assumes CH: D. Mauldin, A representation theorem for the second dual of C[0,1]. Studia Math. 46 (1973), 197?200. Página www do seminário

Página www do seminário