Piotr Koszmider mail: P.Koszmider@impan.pl


PRIMEIRO SEMESTRE DO ANO 2016/17

  Projeto de pesquisa financiado pelo Conselho Nacional de Pesquisa (Brasil)

Título do projeto: Topologia analítica e consistencia em estruturas induzidas combinatoriamente

Pesquisador visitante especial - 04.2014-03.2017
Coordenadora: Christina Brech
Página do projeto



Koszmider, Piotr; On the problem of compact totally disconnected reflection of nonmetrizability. Topology Appl. 213 (2016), 154–166
We construct a ZFC example of a nonmetrizable compact space K such that every totally disconnected closed subspace L of K is metrizable. In fact, the construction can be arranged so that every nonmetrizable compact subspace may be of fixed big dimension. Then we focus on the problem if a nonmetrizable compact space K must have a closed subspace with a nonmetrizable totally disconnected continuous image. This question has several links with the structure of the Banach space C(K), for example, by Holsztyński's theorem, if K is a counterexample, then C(K) contains no isometric copy of a nonseparable Banach space C(L) for L totally disconnected. We show that in the literature there are diverse consistent counterexamples, most eliminated by Martin's axiom and the negation of the continuum hypothesis, but some consistent with it. We analyze the above problem for a particular class of spaces. OCA+MA however, implies the nonexistence of any counterexample in this class but the existence of some other absolute example remains open.

Piotr Koszmider, Uncountable equilateral sets in Banach spaces of the form C(K)
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 uncountable equilateral set (or equivalently no uncountable (1+ε)-separated set in the unit sphere for any ε>0) making another consistent combinatorial assumption. The compact K is a version of the split interval obtained from a sequence of functions which behave in an anti-Ramsey manner. It remains open if there is an absolute example of a nonseparable Banach space of the form different than C(K) which has no uncountable equilateral set. It follows from the results of S. Mercourakis, G. Vassiliadis that our example has an equivalent renorming in which it has an uncountable equilateral set. It remains open if there are consistent examples which have no uncountable equilateral sets in any equivalent renorming but it follows from the results of S. Todorcevic that it is consistent that every nonseparable Banach space has an equivalent renorming in which it has an uncountable equilateral set.


Accepted to Isreal Journal of Mathematics


Saeed Ghasemi, Piotr Koszmider; An extension of compact operators by compact operators with no nontrivial multipliers
We construct an essential extension of K(l2(c)) by K(l2), where c denotes the cardinality of continuum, i.e., a C*-subalgebra A of B(l2) satisfying the short exact sequence
0→K(l2)→iA→K(l2(c))→0,
where i[K(l2)] is an essential ideal of A such that the algebra of multipliers M(A) of A is equal to the unitization of A. In particular A is not stable which sheds light on permanence properties of the stability in the nonseparable setting. Namely, an extension of a nonseparable algebra of compact operators, even by K(l2), does not have to be stable. This construction can be considered as a noncommutative version of Mrówka's Ψ-space; a space whose one point compactification equals to its Cech-Stone compactification and is induced by a special uncountable family of almost disjoint subsets of N. The role of the almost disjoint family is played by an almost orthogonal family of projections in B(l2), but the almost matrix units corresponding to the matrix units in K(l2(c)) must be constructed with extra care.

Saeed Ghasemi, Piotr Koszmider, Noncommutative Cantor-Bendixson derivatives and scattered C*-algebras
We analyze the sequence obtained by consecutive applications of the Cantor-Bendixson derivative for a noncommutative scattered C*-algebra A, using the ideal IAt(A) generated by the minimal projections of A. With its help, we present some fundamental results concerning scattered C*-algebras, in a manner parallel to the commutative case of scattered compact or locally compact Hausdorff spaces and superatomic Boolean algebras. It also allows us to formulate problems which have motivated the "cardinal sequences" programme in the classical topology, in the noncommutative context.
This leads to some new constructions of noncommutative scattered C*-algebras and new open problems. In particular, we construct a type I C*-algebra which is the inductive limit of stable ideals Aα, along an uncountable limit ordinal λ, such that Aα+1/Aα is *-isomorphic to the algebra of all compact operators on a separable Hilbert space and Aα+1 is σ-unital and stable for each α<λ, but A is not stable and where all ideals of A are of the form Aα. In particular, A is a nonseparable C*-algebra with no ideal which is maximal among the stable ideals. This answers a question of M. Rordam in the nonseparable case. All the above C*-algebras Aαs and A satisfy the following version of the definition of an AF algebra: any finite subset can be approximated from a finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers.

Antonio Aviles, Piotr Koszmider, A 1-separably injective space that does not contain l
We study the ω2-subsets of tightly σ-filtered Boolean algebras and, as an application, we show the consistency of the existence of a Banach space that is 1-separably injective but does not contain any isomorphic copy of l.

Tristan Bice, Piotr Koszmider, A note on the Akemann-Doner and Farah-Wofsey constructions
We remove the assumption of the continuum hypothesis from the Akemann-Doner construction of a non-separable C*-algebra A with only separable commutative C*-subalgebras. We also extend a result of Farah and Wofsey's, constructing ℵ1 commuting projections in the Calkin algebra with no commutative lifting. This removes the assumption of the continuum hypothesis from a version of a result of Anderson. Both results are based on Luzin's almost disjoint family construction.


Accepted to Proceedings of the AMS


  Orientaçao de doutorado

Damian Sobota. Instituto de Matemática da Academia Polonesa de Ciencias

O título da tese: Cardinal invariants of the continuum and convergence of measures on compact spaces

Defesa com distin¸ão: 19.10.2016.



Orientaçao de doutorado
Clayton Suguia Hido. Universidade de São Paulo; Co-orientado com Prof. Christina Brech.

O tema da tese: Métodos conjuntistas em análise funcional

Data da defesa aproximada: 2018

Visita científica em Varsóvia: 01.04- 31.12.2016


Estagiário de pósdoutorado

Saeed Ghasemi 09.2015 - 09.2016. Estagiário de pósdoutorado, no Instituto de Matemática da Academia Polonesa de Ciencias


  • Métodos conjuntistas em C*-álgebras

Estagiário de pósdoutorado

Tristan Bice 09.2016 - 09.2018. Estagiário de pósdoutorado, no Instituto de Matemática da Academia Polonesa de Ciencias


  • Métodos conjuntistas em C*-álgebras

Organizaçao de Seminário do IMPAN: Grupo de trabalho em aplicaçoes de teoria dos conjuntos
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


Membro de comissoes

Membro da comissao de programa de doutorado de IM PAN

Member da comisao de WCMCS sobre assuntos de pesquisadores jovens


Visita científica
Visita científica na Universidade de São Paulo



Trabalho editorial
Membro da banca editorial Extracta Mathematicae

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


Seminário: Grupo de trabalho em aplicaçoes de teoria dos conjuntos
Palestra no seminario: September 27, 2016, 15.15-17, room 106,

Title: "A nonseparable scattered C*-algebra without a nonseparable commutative subalgebra"


Página www do seminário



Seminário de análise funcional do IMPAN


Seminário de teoria dos conjuntos da Universidade de Varsóvia

Topology Seminar at the University of Warsaw


"O mundo não fica deitado prostrado, esperando para uma ordem e coerência pela generosidade da mente humana. As coisas são evocativas. Quando as presunções estão silenciosas e todas as palavras ficam quietas, o mundo fala. É preciso queimar os clichês para limpar o ar para escutar. Clichês conceituais são falsificações; as noções preconcebidas são os deslocados. Conhecimento inclui amor, zelo pela coisa que buscamos conhecer, saudade, atração, estado de ser esmagado."

    --Abraham Joshua Heschel --