Previous talks this semester:
21.05.2015 at 11.15 room 105.
Marcin Sabok (McGill/IM PAN): On the nonseparability of the space of ndimensional 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 329363
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 1Plichko
Abstract: I will present a recent result of M. Bohata, J. Hamhalter and O. Kalenda
that predual of a von Neumann algebra is 1Plichko, i.e., it has a countably
1norming Markushevich basis (equivalently, there is a commutative
projectional skeleton consisting of norm one projections). In particular, every dual of a C* algebra is 1Plichko.
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_{∞}/c_{0}
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_{∞}/c_{0} 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 201415.
Talks in the second semester of 201314.
Talks in the first semester of 201314.
Talks in the second semester of 201213.
Talks in the first semester of 201213.
Talks in the second semester of 201112.
Talks in the first semester of 201112.
