Existence of discrete examples separating C*-uniqueness and *-regularity

Determine whether there exist discrete groups G such that (i) G is amenable but L1(G) is not C*-unique (i.e., admits more than one C*-norm), or (ii) L1(G) is C*-unique while G is not *- regular.

Background

Boidol exhibited (non-discrete) amenable groups that are not C*-unique and groups that are C*-unique but not *-regular, showing strict containment between these classes. Whether analogous examples exist in the discrete setting is explicitly stated as unknown and motivates further questions about uniqueness and regularity for discrete amenable groups.

References

Boidol gave examples of amenable groups which are not C *- unique, and of C *- unique groups which are not *- regular [Boi84, p.230], thereby showing strict con- tainment between these classes of groups. However, it remains unknown if such examples exist among discrete groups, prompting Leung and Ng to pose the ques- tion if all discrete, amenable group are C *- unique [LN04].

— The ideal separation property for reduced group $C^*$-algebras (2408.14880 - Austad et al., 27 Aug 2024) in Introduction