Homologically trivial Frechet algebras

We survey some results on the structure of biprojective, contractible and superbiprojective Frechet algebras. Also, we characterize biprojective Frechet algebras in terms of derivations and multipliers, and we obtain a new proof of Helemskii's theorem on the vanishing of the three-dimensional cohomology groups of biprojective algebras. Finally, we describe biprojective and superbiprojective sigma-C*-algebras. In particular, we show that each biprojective sigma-C*-algebra is topologically *-isomorphic to the cartesian product of a countable family of biprojective C*-algebras.