Dans ce travail, nous étudions le problème de décomposicion suivant: Étant donnés deux ouverts bornés de Cp, Ω1 et Ω2 (vérifiant certaines conditions) et étant donnée une matrice A(z), carrée d'ordre n, dont les coefficients sont des fonctions holomorphes dans Ω1 ∩ Ω2, ayant une prolongement C∞ à l'adhérence (Ω1 ∩ Ω2), peut-on trouver deux matrices A1(z), A2(z) holomorphes dans Ω1 et Ω2 respectivement et se prolongeant de manière C∞ à (Ω1) et (Ω2) telles que sur Ω1 ∩ Ω2 on aitA = A1A2.

Let A be a Banach algebra, and let d: A → A be a continuous derivation such that each element in the range of d has a finite spectrum. In a series of papers it has been proved that such a derivation is an inner derivation implemented by an element from the socle modulo the radical of A (a precise formulation of this statement can be found in the Introduction). The aim of this paper is twofold: we extend this result to the case where d is not necessarily continuous, and we give a complete description...

Soient $A$ une algèbre de Banach complexe, $G{L}_{\infty }\left(A\right)$ le groupe général linéaire stable de $A$ et $G{L}_{\infty }^0\left(A\right)$ sa composante connexe pour la topologie normique. Nous montrons que toute trace non nulle $r:A\to \mathbf{C}$ permet de définir un homomorphisme ${\Delta }_r$ de $G{L}_{\infty }^0\left(A\right)$ sur le quotient du groupe additif $\mathbf{C}$ par l’image $\underset{\_}{r}(K_0\left(A\right))$ du groupe de Grothendieck de $A$. Si $A=M_n\left(\mathbf{C}\right)$ (respectivement si $A$ est un facteur fini continu) avec la trace usuelle, alors $\exp \left(i2\pi {\Delta }_r\right)$ est le déterminant usuel (resp. $\exp \left(\mathrm{Re}\right(i2\pi {\Delta }_r\left)\right)$ est celui de Fuglede et Kadison). Dans le cas général, les déterminants ${\Delta }_r$ permettent...

Entire functions operate in complete locally A-convex algebras but not continuously. Actually squaring is not always continuous. The counterexample we give is multiplier algebra.

We recall from [9] the definition and properties of an algebra cone C of a real or complex Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A. The Banach algebra A is then called an ordered Banach algebra. An important property that the algebra cone C may have is that of normality. If C is normal, then the order structure and the topology of A are reconciled in a certain way. Ordered Banach algebras have interesting spectral properties....

We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way to C*- and W*-algebras; we show that interpolation space techniques can be used in place of GNS type arguments....

We deal with dual complementors on complemented topological (non-normed) algebras and give some characterizations of a dual pair of complementors for some classes of complemented topological algebras. The study of dual complementors shows their deep connection with dual algebras. In particular, we refer to Hausdorff annihilator locally C*-algebras and to proper Hausdorff orthocomplemented locally convex H*-algebras. These algebras admit, by their nature, the same type of dual pair of complementors....