Seminar: Working group in applications of set theory, IMPAN

Thursday, 29.11.2018, 10:15, room 105, IMPAN

Speaker: Piotr Koszmider (IM PAN)

Title: “The Grothendieck property for Banach spaces of continuous functions”

Abstact: “In the first talk of the series devoted to classical phenomena in Banach spaces of the form C(K) we will see how weakly compact sets in the dual space to C(K) generalize finite subsets of K. The concrete goal will be to motivate the Grothendieck property (weak and weak* convergence of sequences coincide in the dual) for C(K)s as a generalization of K having no nontrivial convergent sequence and to prove that l∞≡C(βN) has the Grothendieck property. This will require the proof of the Grothendieck-Dieudonne characterization of weakly compact sets in the spaces of measures. All the results and proofs presented during the talk are in classical texbooks, but we will try to represent combinatorial and topological bias, leading in the following talks to more set-theoreic issues. The purpose of this series of talks is to introduce particpants with the set-theoretic topological background to some topics related to Banach spaces of the form C(K). The area is quite sensitive to infinitary combinatorics, e.g., Talagrand: CH implies that there is infinite K such that C(K) is Grothendieck but does not have l∞ as its quotient; Haydon, Levy, Odell: p=2^ω>ω_1 implies that every Grothendieck C(K) for K infinite has l∞ as its quotient “.

Visit our seminar page which may include some future talks at https://www.impan.pl/~set_theory/Seminar/