On the operator homology of the Fourier algebra and its cb-multiplier completion

We study various operator homological properties of the Fourier algebra A(G) of a locally compact group G. Establishing the converse of two results of Ruan and Xu [35], we show that A(G) is relatively operator 1-projective if and only if G is IN, and that A(G) is relatively operator 1-flat if and only if G is inner amenable. We also exhibit the first known class of groups for which A(G) is not relatively operator C-flat for any C≥1. As applications of our techniques, we establish a hereditary property of inner amenability, answer an open question of Lau and Paterson [24], and answer an open question of Anantharaman-Delaroche [1] on the equivalence of inner amenability and Property (W). In the bimodule setting, we show that relative operator 1-biflatness of A(G) is equivalent to the existence of a contractive approximate indicator for the diagonal GΔ in the Fourier-Stieltjes algebra B(G×G), thereby establishing the converse to a result of Aristov, Runde, and Spronk [3]. We conjecture that relative 1-biflatness of A(G) is equivalent to the existence of a quasi-central bounded approximate identity in L1(G), that is, G is QSIN, and verify the conjecture in many special cases. We finish with an application to the operator homology of Acb(G), giving examples of weakly amenable groups for which Acb(G) is not operator amenable.