impan seminar:

working group in applications of set theory



June 12th, Tomasz Kochanek (IMPAN) 1115 room 105


Bourgain's theorem characterizing non-Banach-Saks operators on C(K)-spaces
I will present the proof (or, at least, a detailed sketch of such) of Bourgain's theorem saying that a bounded linear operator T: C(K) --> X fails to have the Banach-Saks property if and only it fixes (acts as an isomorphism on) some copy of C(ωω+). The two main tools are: Alspach's theorem about equivalence between fixing a copy of C(ωω+) and the inequality Sz(T) >= ω2 for the Szlenk index of T, and Rosenthal's dichotomy theorem about Cesaro summable sequences and spreading models isomorphic to l1. I shall also explain the key role which is played here by C(ωω), by recalling Schreier's proof of the fact that this space fails the Banach-Saks property.

Previous talks this semester:
May 8th, Przemysław Ohrysko (IMPAN) 1115 room 105: On the spectra of measures
Abstract: Since the set of measures with natural spectrum is not closed under addition, I will introduce new closed *subalgebra of M(T) which consists of "spectrally reasonable measures". Next, I will show that Zafran's ideal belong to this class. There are many intresting open questions related to this topic - for example: are discrete measures reasonable? The notes of the talk can be found here. The paper the talk is based on can be found here

April 10, 2014 Lectures of Michael Megrelishvili

March 27, 2014, 1115, room 105: Cristobal Rodriguez (Paris 7): Fragments of the paper "On isomorphisms of Banach spaces of continuous functions" (To appear in Israel J. Math) by G. Plebanek will be presented. The paper contains new insights into the structure of the above isomorphisms and embeddings of C(K)s.

March 20, 2014, 1115, room 408: Leandro Candido (USP/IM PAN): A Banach-Stone type theorem by Cambern will be presented with the proof. It says that if T is an ismorphism between C(K) and C(L) such that ||T||||T-1||<2, then K and L are homeomorphic.

March 6, 13. 2014, 1115, room 105, Damian Sobota (WCMCS/PWr)

During the next two meetings of the seminar I would like to present the relations between the Grothendieck property of Banach spaces and the Nikodym property of non-sigma-complete Boolean algebras. A Banach space X has the Grothendieck property (G) if every weak* null sequence in X* is weakly null. A Boolean algebra B has the Nikodym property (N) if every pointwise bounded sequence of finitely additive measures on B is uniformly bounded.

The plan is as follows. First, I will show what properties of Banach spaces are implied by (G) and what the most interesting examples of such spaces are. Especially, I focus on Banach spaces C(K) of continuous functions over some compact Hausdorff space K. The spaces with (G) in such a case naturally appear when K is a Stone space of a Boolean algebra with some additional property. E.g. the Grothendieck theorem states that C(K) has (G) when K is extremely disconnected, i.e. the Boolean algebra Clopen(K) of clopen subsets of K is complete. In case Clopen(K) is not complete, for C(K) to be Grothendieck we need some separation or interpolation properties of Clopen(K), e.g. Haydon's Subsequential Separation Property or Freniche's Subsequential Interpolation Property.

During the second lecture I will talk why the Nikodym property of Boolean algebras is important. I will show what separation or completeness properties of Boolean algebras cause them having (N). Finally, I plan to prove that the algebra of Jordan-measurable subsets of [0,1] has (N) but its C(K) lacks (G) (the result due to Schachermayer). This was the first example of such an algebra; now thanks to Graves and Wheeler we know more of them. However little is still known in the case of the opposite issue, i.e. whether there exists a Boolean algebra without (N) but such that its C(K) has (G) -- the only positive result in this matter is a difficult construction under CH due to Talagrand.










Talks in the first semester of 2013-14.

Talks in the second semester of 2012-13.

Talks in the first semester of 2012-13.

Talks in the second semester of 2011-12.

Talks in the first semester of 2011-12.

Time and place: Thursdays 11-13 am, room 105, Sniadeckich 8

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The scope of the seminar: Set-theoretic combinatorial and topological methods in diverse fields of mathematics, with a special emphasis on abstract analysis like Banach spaces, Banach algebras, C*-algebras, Here we include both the developing of such methods as forcing, descriptive set theory, Ramsey theory as well as their concrete applications in the fields mentioned above.

Working group style: We will make efforts so that this seminar has more a working character rather than the presentation style. This means that we encourage long digressions, discussions, background preparations and participation of everyone. To achive this, the duration of the seminar will not be rigidly fixed and may reach even 3 hours with the breaks. We would like to immerse ourselves into the details of the mathematical arguments studied.

Participants this semester so far:

  • Leandro Candido (IM PAN/USP)
  • Michal Doucha (IMPACT/IM PAN)
  • Tomasz Kochanek (IM PAN)
  • Piotr Koszmider (IM PAN)
  • Mikłoaj Krupski (Ph. D. student IM PAN)
  • Przemysław Ohrysko (Ph. D. student IM PAN)
  • Cristóbal Rodriguez (Ph. D. student Paris 7)
  • Damian Sobota (Ph. D. student WCMCS/PWr)
  • Michał Wojciechowski (IM PAN)
Forthcoming talks:

  • No more talks this semester.