Classification of B-ideals representable in countable C(K) spaces

Characterize which B-ideals can be represented in C(K) for a countable compact space K.

Background

Theorem 6.19 proves that every tall B-ideal representable in C(K), for K countable, contains a non-pathological front ideal. The authors leave open a full characterization of the B-ideals admitting such representations.

References

Question 6.20. Which B-ideals can be represented in C(K) for a countable compactum K?

— $F_σ$-ideals, colorings, and representation in Banach spaces  (2501.15643 - Lopez-Abad et al., 26 Jan 2025) in Question 6.20, Section 6.3, page 40