Sobczyk's theorem is usually stated as: every copy of c0 inside a separable Banach space is complemented by a projection with norm at most 2. Nevertheless, our understanding is not complete until we also recall: and c0 is not complemented in l∞. Now the limits of the phenomenon are set: although c0 is complemented in separable superspaces, it is not necessarily complemented in a non-separable superspace, such as l∞.

We investigate some homological notions of Banach algebras. In particular, for a locally compact group G we characterize the most important properties of G in terms of some homological properties of certain Banach algebras related to this group. Finally, we use these results to study generalized biflatness and biprojectivity of certain products of Segal algebras on G.

We will show that for each sequence of quasinormable Fréchet spaces ${\left(E_n\right)}_{\mathbb{N}}$ there is a Köthe space λ such that $Ex{t}^1(\lambda \left(A\right),\lambda \left(A\right)=Ex{t}^1(\lambda \left(A\right),E_n)=0$ and there are exact sequences of the form $...\to \lambda \left(A\right)\to \lambda \left(A\right)\to \lambda \left(A\right)\to \lambda \left(A\right)\to E_n\to 0$. If, for a fixed ℕ, $E_n$ is nuclear or a Köthe sequence space, the resolution above may be reduced to a short exact sequence of the form $0\to \lambda \left(A\right)\to \lambda \left(A\right)\to E_n\to 0$. The result has some applications in the theory of the functor $Ex{t}^1$ in various categories of Fréchet spaces by providing a substitute for non-existing projective resolutions.

We study the structure of certain classes of homologically trivial locally C*-algebras. These include algebras with projective irreducible Hermitian A-modules, biprojective algebras, and superbiprojective algebras. We prove that, if A is a locally C*-algebra, then all irreducible Hermitian A-modules are projective if and only if A is a direct topological sum of elementary C*-algebras. This is also equivalent to A being an annihilator (dual, complemented, left quasi-complemented, or topologically...

Soient ${𝔑}_n$ (resp. ${ℰ}_n$) l’anneau des germes de fonctions de Nash (resp. l’anneau des germes de fonctions $C^{\infty }$) à l’origine de ${\mathbf{R}}^n$: ${ℬ}_n$ (resp. ${ℬ}_n^{\text{'}}$) le module sur ${𝔑}_n$ des germes de fonctions de Bernstein $C^{\infty }$ (resp. le module sur ${𝔑}_n$ des germes de distributions de Bernstein) à l’origine de ${\mathbf{R}}^n$. Les deux résultats principaux de l’article sont les suivants : ${ℬ}_n^{\text{'}}$ est un module injectif sur ${𝔑}_n$ et ${ℰ}_n/{ℬ}_n$ est un module plat sur ${𝔑}_n$.