impan seminar:

working group in applications of set theory

No more talks this academic year
Previous talks this semester:

Exceptional time and room: 12.06.2015 at 9.30 room 106.

Michal Doucha (IMPACT/IM PAN): On group C*-algebras

Abstract: I will try to give an idea why people study group C*-algebras and what the connection between them and unitary group representations is. I will present two main constructions

  • the construction of a reduced group C*-algebra
  • the construction of a full group C*-algebra
One of rather easy but interesting observations will be that the full group algebra C*(F_infty) of the free group of countably many generators is a projectively universal separable C*-algebra; i.e. for every separable C*-algebra B there is a *-morphism from C*(F_infty) onto B. Then depending on the interest, I can try to show that in the case of amenable groups these two constructions lead to the same object. I will focus mostly on discrete groups but I will sketch how everything works as well for non-discrete locally compact groups.

21.05.2015 at 11.15 room 105.

Marcin Sabok (McGill/IM PAN): On the non-separability of the space of n-dimensional operator spaces for n>2

Abstract: We will discuss the argument of Junge and Pisier from the article: Bilinear forms on exact operator spaces and B(H)⊗B(H), Geometric & Functional Analysis GAFA, 1995, Volume 5, Issue 2, pp 329-363

Additional comment: This result implies that unlike in the world of commutative sets where the Cantor set Δ continuously maps onto all metrizable compact spaces which yields that all separable C(K)s embed into C(Δ), in the noncommutative world there is no universal separable C*-algebra.

14.05.2015 at 11.15 room 105.

Marek Cuth (IM PAN/ WCNM): The predual of a von Neumann algebra is 1-Plichko

Abstract: I will present a recent result of M. Bohata, J. Hamhalter and O. Kalenda that predual of a von Neumann algebra is 1-Plichko, i.e., it has a countably 1-norming Markushevich basis (equivalently, there is a commutative projectional skeleton consisting of norm one projections). In particular, every dual of a C* algebra is 1-Plichko. I will concentrate on the techniques coming from the theory of von Neumann algebras and their applications contained in the proof.

07.05.2015 at 11.15 room 105.

Piotr Koszmider (IM PAN): Traces of operators and the set theory of the Banach space C(N*)

Abstract: A trace on a Banach algebra A is a linear bounded functional τ on A such that τ(ab)=τ(ba) for all a, b in A. A charater is a linear bounded functional τ on A such that τ(ab)=τ(a)τ(b) for all a, b in A. We will review known facts and open questions concerning the existence of nonzero characters or nonzero traces on the algebra B(X) of all linear bounded operators on a Banach space X. For example, generalizing P. Halmos' result concerning operators on the Hilbert space N. Laustsen showed that if X is "infinitely divisible", then every operator on X is a sum of commutators and so no nontrivial trace can exist on B(X). The continuum hypothesis implies that the Banach space C(N*)≡ l/c0 is "infinitely divisible" (Negrepontis), but we do not know if this is the case in ZFC. For example in the Cohen model C(N*) is not isomorphic to any l-sum of Banach spaces (C. Brech, P. Koszmider). So l/c0 may provide a consistent negative answer to a question of A. Villena whether for every Banach space X isomorphic to its square, the algebra B(X) has no nontrivial traces.

23.04.2015 at 11.15 room 105.

Marek Cuth (IM PAN): Do countable models of ZFC generate club of separable subspaces in a Banach space?

Abstract: First, I will briefly talk about separable reductions and try to explain why it is interesting to solve the problem from the title. Then, I will talk about some partial results and about the difficulities when trying to solve the problem in full generality. Separable reduction is a method of extension the validity of a statement from separable spaces to the nonseparable setting not involving the proof of the statement in the separable case. This method is based on a construction of a separable subspace with certain properties. One possibility of constructing a separable subspace of a Banach space X is to take countable model of ZFC (more precisely, of a finite fragment of ZFC), call it M, and then take the closure of the points from M which are in X. The question is, how good the family of so constructed separable subspaces can be. Namely, is it possible to find a family of countable models of ZFC in such a way that those subspaces will form a club?

Alternative activities

Talks in the first semester of 2014-15.

Talks in the second semester of 2013-14.

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 11.15-13.00 am, room 105, Sniadeckich 8

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. We would like to immerse ourselves into the details of the mathematical arguments studied. Also the talks are usualy devoted to research in progress or fascinating results leading to some project not yet resolved. While ready final results could be presented at other seminars at IM PAN or UW.

Participants this semester so far:

  • Marek Cuth (WCMCS-IM PAN)
  • Michal Doucha (IMPACT/IM PAN)
  • Mehrdad Kalantar (IM PAN)
  • Piotr Koszmider (IM PAN)
  • Adam Krawczyk (MIM UW)
  • Gabriel Salazar (IM PAN)
  • Marcin Sabok (McGill/IMPAN)
  • Damian Sobota (Ph. D. student IM PAN)
  • Michał Świętek (Ph. D. student UJ/WCMCS-IM PAN)
Forthcoming talks:

  • No more seminars this academic year