Largest type I ideal generated by finite unions of sparse subsets

Determine whether the ideal obtained as the closed union of the compact ideals I_A associated with subsets A of a countable group Γ that are finite unions of sparse subsets is the largest type I ideal in the uniform Roe algebra C_u^*(Γ).

Background

For a subset A of a countable group Γ, the paper defines the compact ideal I_A in the uniform Roe algebra C_u*(Γ). It proves that if A is a finite union of sparse subsets, then I_A is type I. Since finite unions of sparse subsets are directed under inclusion, the closed union of the corresponding ideals forms a type I ideal.

The paper also proves that, for exact groups, the largest type I ideal is contained in the largest AF ideal. However, it does not establish whether the type I ideal constructed from finite unions of sparse subsets exhausts the largest type I ideal. This leaves open a precise comparison between the geometric construction and the abstract largest type I ideal.

References

Clearly, a union of two sets that are finite unions of sparse sets is still a finite union of sparse sets, and thus the family of subsets of $\Gamma$ that are finite unions of sparse sets, is directed by inclusion. Hence, the closed union of compact ideals associated to such subsets of $\Gamma$ is an ideal in $C_u*(\Gamma)$ of type I. It remains open if this ideal is the largest type I ideal in general.

— On AF- and type I-ideals in certain crossed product C$^\ast$-algebras  (2609.11297 - Ravnanger, 10 Sep 2026) in Section “Uniform Roe algebras”, paragraph immediately preceding Proposition on directed unions of type I ideals