Let \(G\) be a locally compact group, and let \(L^1(G)\) be its group algebra. Then the following Banach spaces are left Banach \(L^1(G)\)-modules in a canonical manner: \(L^1(G)\), \(C_0(G)\), \(L^\infty(G)\), \(L^1(G)''\), and \(L^p(G)\) for \(p \in (1,\infty)\).
In the paper under review, the authors systematically investigate for which \(G\) any of the modules above satisfy the following properties that arise canonically in A. Ya. Helemskĭi’s topological homology [The homology of Banach and topological algebras (Dordrecht etc. 1989; Zbl 0695.46033)]: projectivity, injectivity, and flatness.
In some cases, a complete characterization is possible: for instance, \(L^1(G)\) is projective for every \(G\) whereas \(M(G)\) is projective if and only if \(G\) is discrete.
In other cases, the authors only obtain partial characterizations: if \(L^1(G)''\) is projective, then \(G\) is discrete and contains no infinite, amenable subgroup. (They conjecture that \(L^1(G)''\) is projective if and only if \(G\) is finite.)
The discussion for which \(G\) the module \(L^p(G)\) with \(p \in (1,\infty)\) is injective is particularly interesting. Since \(L^p(G)\) is a dual module, it is clear that it is injective if \(G\) is amenable. On the other hand, \(L^p(G)\) is not injective if \(G\) is the free group in two generators. The authors believe, but are unable to prove, that, for discrete \(G\) at least, \(\ell^p(G)\) is injective if and only if \(G\) is amenable. They obtain a partial confirmation of this conjecture through the notion of pseudo-amenability: this is a technical condition for discrete groups that is implied by amenability, and that, in turn, implies that the group does not contain the free group in two generators. They show that, if \(\ell^p(G)\) is injective, then \(G\) is pseudo-amenable.