Damian Sobota: Rosenthal families and ultrafilters

Dear all,

The seminar meets on Wednesday May 23rd at 11:00 in the Institute of
Mathematics CAS, Zitna 25, seminar room, 3rd floor, front building.

Program: Damian Sobota — Rosenthal families and ultrafilters

Rosenthal’s lemma is a simple technical result with numerous
applications to measure theory and Banach space theory. The lemma in its
simplest form reads as follows: “For every infinite real-entried matrix
(m(n,k): n,k in N) such that every entry is non-negative and the sum of
every row is <=1, and every epsilon>0, there exists an infinite subset A
of N such that for every k in A we have sum_{n in A, n\neq
k}m_n^k<epsilon.” A natural question arises whether we can choose the
set A from a previously fixed family F of infinite subsets of N. If F
has such a property, then we call it Rosenthal. Thus, Rosenthal’s lemma
states that [N]^omega is Rosenthal. During my talk I’ll present some
necessary or sufficient conditions for a family to be Rosenthal and
prove that under MA(sigma-centered) there exists a P-point which is a
Rosenthal family but not a Q-point. (No Banach space will appear during
the talk.)

Best,
David