impan seminar:

working group in applications of set theory


Previous talks this semester:

23.01.2014 at 11.15, room 105; Przemysław Ohrysko, On the limits of Fourier coefficents along ultrafilters

Abstract. It will be shown that there are two distinct ultrafilters on the integers u and v such that the u-limits and v-limits of Fourier coeffcients are the same. We know from different considerations that the Gelfand space of the measure algebra with convolutions contains βN. The above shows that nevertheless the most natural map from βN into the Gelfand space is not injective.

09.01, 16.01 at 11.15, room 105; Tomasz Kochanek, On the largest regular subalgebra of the measure algebra on a locally compact Abelian group

This talk is meant to be a continuation/extension of the recent talk by P. Ohrysko on the spectrum of the Banach algebra M(T) of all bounded regular complex Borel measures on the unit circle.
First, we discuss basic properties of the hull-kernel topology on Gelfand space, as well as the notion of regular algebra. From the Wiener-Pitt phenomenon we deduce that for every non-discrete locally compact Abelian group G the measure algebra M(G) is not regular, and hence its spectrum cannot be totally disconnected (tha fact we have already observed during the talk by P. Ohrysko). The same will be proved about the Banach algebras L_1(R+) and l_1(Z+).
In the second part of the talk we focus on the existence and properties of the largest regular subalgebra Reg(A) of a given Banach algebra A, in particular, in the case A = M(G). Although no measure-theoretic characterisation of Reg(M(G)) is known, we shall show that it is quite a huge subalgebra of M(G) containing, for instance, all absolutely continuous and all discrete measures. We also identify the spectrum of Reg(M(G)) as a Hausdorff compactification of the dual group of G that does not coincide with the Bohr compactification, unless G is discrete. Finally, we discuss the relation between Reg(A) and the Apostol algebra of A.

5.12, 12.12 at 11.15 am. room 105; Adam Kwela, on the paper " G. Horbaczewska, A. Skalski; The Banach principle for ideal convergence in the classical and noncommutative context. J. Math. Anal. Appl. 342 (2008), no. 2, 1332–1341.

21.11, 28.11 - Piotr Koszmider, On certain Banach space

In my talk I willl present some random version of the split interval K and will describe some properities of the Banach space C(K). It is the space of the paper "P. Koszmider, On a problem of Rolewicz about Banach spaces that admit support sets; J. Funct. Anal. 257 (2009), no. 9, 2723–2741. I hope the anti-Ramsey theoretic properties of the space can be exploited much more.

24.10, 31.10 - Przemysław Ohrysko, On the Arens - Royden theorem. This theorem implies that the space of multiplicative funtionals on M(T) with the weak* topology is not homeomorphic to βN.

The participants should revise the following topics before the talks:

  • Basics:
    1. Banach algebras, spectrum and spectral radius.
    2. Gelfand theory, Gelfand transform, semisimplicity, multiplicative linear functionals, functional calculus.
    3. Information about structure of the group of invertible elements as in book "Banach algebras" by W. Zelazko or in book "Functional Analysis" by W. Rudin.
  • Additional definitions and theorems related to them:
    1. join spectrum of elements in commutative Banach algebras.
    2. multivariable functional calculus.
    3. Shilov idempotent theorem.

10.10.2013 and 17.10.2013. at 1pm, room 106

Tomasz Kochanek, Twisted sums with C(K)-spaces

Based on F. Cabello Sanchez, J.M.F. Castillo, N.J. Kalton, D.T. Yost, Twisted sums with C(K) spaces, TAMS 355 (2003), 4523-4541;


3.10.2013. at 1pm, room 106

Cristóbal Rodriguez (Paris 7); The Open Coloring Axiom and embeddings into P(N)/fin.

Abstract; The Open Coloring Axiom (OCA), as introduced by Todorcevic in 1989, is a combinatorial principle that has been very successful in answering questions related to the automorphism group of P(N)/fin, and has found applications in other structures such as C*-algebras. In this session we intend to introduce the axiom, showing some easy applications and stating some important consequences, and then we will go on to show a more elaborate use of the principle by sketching the proof of the following theorem by Todorcevic: If I is a nonatomic analytic ideal on N, then P(N)/I is not isomorphic to a subalgebra of P(N)/fin.

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 at 1pm, room 106, Sniadeckich 8

Sometimes the seminar will take place on at 11 am.

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:

  • Michal Doucha (IMPACT/IM PAN)
  • Christina Brech (USP)
  • Clayton Suguio Hida (M.A. student USP)
  • Tomasz Kania (WCMSC/IMPAN)
  • Krystian Kazaniecki (Ph.D. student UW)
  • Tomasz Kochanek (IM PAN)
  • Piotr Koszmider (IM PAN)
  • Mikłoaj Krupski (Ph. D. student IM PAN)
  • Adam Kwela (Ph. D. student IM PAN)
  • Witold Marciszewski (UW)
  • Przemysław Ohrysko (Ph. D. student IM PAN)
  • Cristóbal Rodriguez (Ph. D. student Paris 7)
  • Adam Skalski (IM PAN)
  • Michał Wojciechowski (IM PAN)
Forthcoming talks:

No more talks this semester. We start March 6.