\(F_\sigma\)-ideals, colorings, and representation in Banach spaces.

Jordi López-Abad (UNED Madrid)

Mar 20. 2025, 09:00 — 09:45

We study \(B\)- and \(C\)-ideals associated with sequences in Banach spaces, where \(C((x_n)n)\) consists of sets for which the series \(\sum_{n\in A} x_n\) is unconditionally convergent, and \(B((x_n)_n)\) consists of sets where the series is weakly unconditionally convergent. We describe these ideals in universal function spaces, particularly in \(C([0,1])\) and \(C(2^{\mathbb N})\), addressing a question by Borodulin-Nadzieja et al.

 

A key aspect is the role of \(c_0\)-saturated spaces and their connection to c-coloring ideals, which exhibit a rich combinatorial structure. We show that for \(d\ge 3\), the random d-homogeneous ideal is pathological, construct hereditarily non-pathological c-coloring ideals, and prove that every \(B\)-ideal in \(C(K)\), for countable \(K\), contains a c-coloring ideal.

 

These results highlight the interplay between combinatorial properties of ideals and their Banach space representations. This is a joint work with V. Olmos and C. Uzcátegui.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Associated Event:
Structures in Banach Spaces (Workshop)
Organizer(s):
Antonio Aviles (U Murcia)
Vera Fischer (U of Vienna)
Grzegorz Plebanek (U of Wroclaw)
Damian Sobota (U of Vienna)