Amenability and co-amenability in non-abelian group duality

Leptin’s theorem asserts that a locally compact group is amenable if and only if its Fourier algebra has a bounded (by one) approximate identity. In the language of locally compact quantum groups—in the sense of J. Kustermans and S. Vaes—, it states that a locally compact group is amenable if and only if its quantum group dual is co-amenable. It is an open problem whether this is true for general locally compact quantum groups. We approach this problem focussing on the rôle of multiplicative unitaries. For a Hilbert space H, a multiplicative unitary W ∈ B(H⊗̃2H) defines a co-multiplication ΓW on B(H), so that (B(H),ΓW ) is a Hopf–von Neumann algebra. We introduce the notion of an admissible, multiplicative unitary. With an admissible, multiplicative unitary W , we associate another Hopf–von Neumann algebra (M ,ΓW ). We show that (B(H),ΓW ) is left amenable (co-amenable) if and only if this is true for (M ,ΓW ). Setting Ŵ := σW ∗σ, where σ is the flip map on H⊗̃2H, we prove that the left co-amenability of (B(H),ΓW ) implies the left amenability of (B(H),Γ Ŵ ), and—for infinite-dimensional H and under an additional technical hypothesis—also establish the converse. Applying these results to locally compact quantum groups—and, in particular, to Kac algebras—, we obtain that a Kac algebra is amenable if and only if its dual is co-amenable. This extends Leptin’s theorem to Kac algebras and answers a problem left open by D. Voiculescu.