2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64226In 1972, B. E. Johnson proved that a locally compact group $G$ is amenable if and only if certain Hochschild cohomology groups of its convolution algebra $L^1(G)$ vanish. Similarly, $G$ is compact if and only if $L^1(G)$ is biprojective: In each case, a classical property of $G$ corresponds to a cohomological propety of $L^1(G)$. Starting with the work of Z.-J. Ruan in 1995, it has become apparent that in the non-commutative setting, i.e. when dealing with the Fourier algebra $A(G)$ or the Fourier-Stieltjes algebra $B(G)$, the canonical operator space structure of the algebras under consideration has to be taken into account: In analogy with Johnson's result, Ruan characterized the amenable locally compact groups $G$ through the vanishing of certain cohomology groups of $A(G)$. In this paper, we give a survey of historical developments, known results, and current open problems.12 pages; a survey article; typos removed, references updatedFunctional AnalysisK-Theory and HomologyOperator Algebras22D15, 22D25, 43A20, 43A30, 46H20 (primary), 46H25, 46L07, 46M18, 46M20, 47B47, 47L25, 47L50text