We study various operator homological properties of the Fourier algebra of a locally compact group G. Establishing the converse of two results of Ruan and Xu [35], we show that is relatively operator 1-projective if and only if G is IN, and that is relatively operator 1-flat if and only if G is inner amenable. We also exhibit the first known class of groups for which is not relatively operator C-flat for any . 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 is equivalent to the existence of a contractive approximate indicator for the diagonal in the Fourier–Stieltjes algebra , thereby establishing the converse to a result of Aristov, Runde, and Spronk [3]. We conjecture that relative 1-biflatness of is equivalent to the existence of a quasi-central bounded approximate identity in , that is, G is QSIN, and verify the conjecture in many special cases. We finish with an application to the operator homology of , giving examples of weakly amenable groups for which is not operator amenable.

Additional Metadata
Keywords Fourier algebra, Group von Neumann algebra, Operator homology
Persistent URL dx.doi.org/10.1016/j.jfa.2017.06.024
Journal Journal of Functional Analysis
Crann, J, & Tanko, Z. (Zsolt). (2017). On the operator homology of the Fourier algebra and its cb-multiplier completion. Journal of Functional Analysis, 273(7), 2521–2545. doi:10.1016/j.jfa.2017.06.024