The theory of quasimultipliers in Banach algebras is developed in order to provide a mechanism for defining the boundary values of analytic semigroups on a sector in the complex plane. Then, some methods are presented for deriving lower estimates for operators defined in terms of quasinilpotent semigroups using techniques from the theory of complex analysis.

We continue our study of topological partial *-algebras, focusing on the interplay between various partial multiplications. The special case of partial *-algebras of operators is examined first, in particular the link between strong and weak multiplications, on one hand, and invariant positive sesquilinear (ips) forms, on the other. Then the analysis is extended to abstract topological partial *-algebras, emphasizing the crucial role played by appropriate bounded elements, called ℳ-bounded. Finally,...

Soient $a_1,...,a_n$ des éléments d’une $b$-algèbre commutative unifère $A$. On définit et étudie un “spectre” de $a=(a_1,...,a_n)$ qui dépend de la croissance des fonctions $u_1\left(s\right),...,u_n\left(s\right)$ de l’égalité spectrale$$(a_1-s_1)u_1\left(s\right)+\cdots +(a_n-s_n)u_n\left(s\right)=1$$près du spectre simultané. À partir des propriétés de ce spectre, on construit un calcul fonctionnel qui, réduit au cas banachique, s’étend à certaines fonctions supposées seulement holomorphes à l’intérieur du spectre simultané. Ce calcul fonctionnel permet aussi d’étudier la régularité des éléments $a_1,...,a_n$ et des fonctions $u_1\left(s\right),...,u_n\left(s\right)$.

It is known that $ℬ\left({\ell }^p\right)$ is not amenable for p = 1,2,∞, but whether or not $ℬ\left({\ell }^p\right)$ is amenable for p ∈ (1,∞) ∖ 2 is an open problem. We show that, if $ℬ\left({\ell }^p\right)$ is amenable for p ∈ (1,∞), then so are ${\ell }^{\infty }\left(ℬ\left({\ell }^p\right)\right)$ and ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$. Moreover, if ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$ is amenable so is ${\ell }^{\infty }(,\left(E\right))$ for any index set and for any infinite-dimensional ${ℒ}^p$-space E; in particular, if ${\ell }^{\infty }\left(\left({\ell }^p\right)\right)$ is amenable for p ∈ (1,∞), then so is ${\ell }^{\infty }\left(\left({\ell }^p\oplus \ell ²\right)\right)$. We show that ${\ell }^{\infty }\left(\left({\ell }^p\oplus \ell ²\right)\right)$ is not amenable for p = 1,∞, but also that our methods fail us if p ∈ (1,∞). Finally, for p ∈ (1,2) and a free ultrafilter over ℕ, we exhibit...