UCT for the finite-dimensional representation completion C*_{fd}(Γ)

Determine whether the C*-algebra C*_{fd}(Γ), obtained by completing the complex group algebra in the direct sum of all finite-dimensional unitary representations, satisfies the Rosenberg–Schochet Universal Coefficient Theorem (UCT), even for a-T-menable groups Γ.

Background

Working with C*{fd}(Γ) would have advantages for representation-approximation problems because it is automatically RFD and directly reflects finite-dimensional representation data. However, the authors emphasize that UCT—a key ingredient in their K-theoretic machinery—may fail or is unverified for C*{fd}(Γ), even when Γ is a-T-menable.

Clarifying UCT for C*_{fd}(Γ) would potentially broaden the applicability of the paper’s stable uniqueness techniques to settings that are more representation-theoretic than the full group C*-algebra.