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, 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 $C\geq1$. As applications of our techniques, we establish a hereditary property of inner amenability, answer an open question of Lau and Paterson, and answer an open question of Anantharaman--Delaroche 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 $G_Δ$ in the Fourier--Stieltjes algebra $B(G\times G)$, thereby establishing the converse to a result of Aristov, Runde, and Spronk. We conjecture that relative $1$-biflatness of $A(G)$ is equivalent to the existence of a quasi-central bounded approximate identity in $L^1(G)$, that is, $G$ is QSIN, and verify the conjecture in many special cases. We finish with an application to the operator homology of $A_{cb}(G)$, giving examples of weakly amenable groups for which $A_{cb}(G)$ is not operator amenable.