March 6. 2014, 1115, room 408,
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.
Time and place: Thursdays 11-13 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. 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 Kania (WCMSC/IMPAN)
- 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)
- March 6, 13: Damian Sobota (WCMCS/PWr) will talk about
the property of Nikodym (N) and the property of Grothendieck (G) of Banach spaces C(K).
While there are C(K)s with N and without G, the only example of C(K) with G and without N
was constructed by
Talagrand assuming CH.
If this can be done without additional set theoretic axioms is unknown.
- March 20: Leandro Candido (USP/IM PAN) will present the proof of a
Banach-Stone type theorem by Cambern.
- April 3, 10: Tomasz Kochanek (IM PAN)
will present a Proof of Bourgain's theorem saying
that if T is an operator from C(K) into a Banach space
which does not have the Banach-Saks property, then it acts as an isomorphism on some copy of C(ωω).