The problem of representability of quadratic functionals by sesquilinear forms is studied in this article in the setting of a module over an algebra that belongs to a certain class of complex Banach *-algebras with an approximate identity. That class includes C*-algebras as well as H*-algebras and their trace classes. Each quadratic functional acting on such a module can be represented by a unique sesquilinear form. That form generally takes values in a larger algebra than the given quadratic functional...

We study the stability of homomorphisms between topological (abelian) groups. Inspired by the "singular" case in the stability of Cauchy's equation and the technique of quasi-linear maps we introduce quasi-homomorphisms between topological groups, that is, maps ω: 𝒢 → ℋ such that ω(0) = 0 and ω(x+y) - ω(x) - ω(y) → 0 (in ℋ) as x,y → 0 in 𝒢. The basic question here is whether ω is approximable by a true homomorphism a in the sense that ω(x)-a(x) → 0 in ℋ as x →...

Soit $K$ un corps commutatif. Chercher une série formelle $S(X,T)\in K\left[\right[X,T\left]\right]$ vérifiant $S(X+Y,T)/S(X,T)\in K\left[\right[Y,T\left]\right]$ conduit naturellement à étudier l’application $U\left(T\right)\to \left(U\right(T){)}^X$, $U\left(T\right)$ étant une unité de l’algèbre $K\left[\right[T\left]\right]$, et à ramener les solutions à la forme $S(X,T)=\sum _{n\ge 0}{H}_n\left(X\right)T^n$, $\left(H_n\right(X\left)\right)$ étant une suite de $K\left[X\right]$ vérifiant les “identités multinomiales” :$$\left(\mu \right)H_n(X_1+...+X_k)=\sum _{{\alpha }_1+...+{\alpha }_k=n}{H}_{{\alpha }_1}(X_1)...H_{{\alpha }_k}(X_k\left)\right(n,k\ge 0).$$Après mise à l’écart par des lemmes combinatoires du cas caract$\left(K\right)\>0$ (les solutions sont triviales), on caractérise de plusieurs manières les solutions. On peut les faire coïncider avec l’ensemble NW des suites de polynômes (ou séries génératrices...