Quotient permanence of C*-uniqueness for discrete groups

Ascertain whether C*-uniqueness of the group algebra L1(G) passes to quotients by closed normal subgroups in the discrete case; specifically, show whether L1(G/N) has a unique C*-norm whenever G is discrete, L1(G) has a unique C*-norm, and N ≤ G is a closed normal subgroup.

Background

While -regularity and amenability are known to pass to certain quotients, the behavior of C-uniqueness under quotients is explicitly noted as unclear, even for discrete groups. Resolving this would clarify permanence properties of uniqueness of the C*-norm for L1(G) under group-theoretic operations.

References

We note that it is not clear if C *- uniqueness passes to quotients by closed, normal subgroups, even in the case of discrete groups; see [LN04, Remark 3.6].

— The ideal separation property for reduced group $C^*$-algebras (2408.14880 - Austad et al., 27 Aug 2024) in Section 3, paragraph preceding Corollary 3.2