Exactness versus C*-exactness for nondiscrete locally compact groups

Determine whether, for nondiscrete locally compact groups G, exactness of G is equivalent to exactness of the reduced group C*-algebra C*_red(G); that is, prove or refute that G is exact if and only if C*_red(G) is an exact C*-algebra.

Background

For discrete groups, exactness of G is known to be equivalent to exactness of C*_red(G). Extending this equivalence to nondiscrete groups is highlighted as a major open problem, with implications for tensor product properties, ideal separation, and structural analysis of reduced group C*-algebras in non-discrete contexts.

References

Conversely, a discrete group G is exact if (and only if) Crea(G) is exact ([KW99, Theorem 5.2]), and it remains a major open problem if this also holds for nondiscrete groups; see [AD02, Problem 9.3] and [Man21].

— The ideal separation property for reduced group $C^*$-algebras (2408.14880 - Austad et al., 27 Aug 2024) in Section 4, paragraph preceding Corollary 4.14