Alaghmandan–Spronk central Fourier algebra amenability conjecture

Alaghmandan and Spronk conjectured that $\mathrm{ZA}(G)$ is amenable exactly when the compact group $G$ is virtually abelian. Their paper was submitted in 2014 and revised in 2015.

Known results

• Virtually abelian $G$: $\mathrm{ZA}(G)$ is amenable (Alaghmandan–Spronk, 2014–2015).
• If $G$ contains a non-abelian closed connected subgroup, $\mathrm{ZA}(G)$ has bounded point derivations and is non-amenable (Alaghmandan–Spronk, 2014–2015).
• For a product of finite groups, $\mathrm{ZA}(G)$ is amenable exactly when all but finitely many factors are abelian (Alaghmandan–Spronk, 2014–2015).
• The identification $A(\operatorname{Conj}(G))\cong\mathrm{ZA}(G)$ is known; a later paper notes gaps in an earlier claimed complete proof.

September 2026 family result

Aleksa Vujičić’s paper claims non-amenability of $\mathrm{ZA}(G)$ for $G=R^d\rtimes\mathrm{GL}_d(R)$, where $R$ is a compact discrete valuation ring, and describes the associated orbit hypergroup. This advances the conjecture for that family only; the claim is unverified here.

Current status (as of September 2026): The conjecture is proved in several substantial cases, including the newly reported family, but remains open in general.