We survey our main results on the density condition for Fréchet spaces and on the dual density condition for (DF)-spaces (cf. Bierstedt and Bonet (1988)) as well as some recent developments.

We introduce two new notions of amenability for a Banach algebra A. The algebra A is n-weakly amenable (for n ∈ ℕ) if the first continuous cohomology group of A with coefficients in the n th dual space $A^{\left(n\right)}$ is zero; i.e., ${\mathscr{H}}^1(A,A^{\left(n\right)})=0$. Further, A is permanently weakly amenable if A is n-weakly amenable for each n ∈ ℕ. We begin by examining the relations between m-weak amenability and n-weak amenability for distinct m,n ∈ ℕ. We then examine when Banach algebras in various classes are n-weakly amenable; we study...

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...

Let E be a Banach space with 1-unconditional basis. Denote by $\Delta \left(\otimes {̂}_{n,\pi }E\right)$ (resp. $\Delta \left(\otimes {̂}_{n,s,\pi }E\right)$) the main diagonal space of the n-fold full (resp. symmetric) projective Banach space tensor product, and denote by $\Delta \left(\otimes {̂}_{n,\left|\pi \right|}E\right)$ (resp. $\Delta \left(\otimes {̂}_{n,s,\left|\pi \right|}E\right)$) the main diagonal space of the n-fold full (resp. symmetric) projective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic, and in addition, that they are isometrically lattice isomorphic to $E_{\left[n\right]}$, the completion of the n-concavification of...

There are given necessary and sufficient conditions on a measure dμ(x)=w(x)dx under which the key estimates for the distribution and rearrangement of the maximal function due to Riesz, Wiener, Herz and Stein are valid. As a consequence, we obtain the equivalence of the Riesz and Wiener inequalities which seems to be new even for the Lebesgue measure. Our main tools are estimates of the distribution of the averaging function f** and a modified version of the Calderón-Zygmund decomposition. Analogous...

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....