Previous talks this semester:
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?
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.