Soient $X$ une variété algébrique complexe, lisse, irréductible, $E$ et $F$ deux espaces vectoriels complexes de dimension finie et $\mu$ un morphisme de $X$ dans l’espace Lin$(E,F)$ des applications linéaires de $E$ dans $F$. Pour $x\in X$, on note $E\left(x\right)$ et $x·E$ le noyau et l’image de $\mu \left(x\right)$, ${\overline{\mu }}_x$ le morphisme de $X$ dans Lin$\left(E\right(x),F/(x·E\left)\right)$ qui associe à $y$ l’application linéaire $v\mapsto \mu \left(y\right)\left(v\right)+x·E$. Soit i${}_{\mu }$ la dimension minimale de $E\left(x\right)$. On dit que $\mu$ ala propriété $\left(𝐑\right)$ en $x$si i${}_{{\overline{\mu }}_x}$ est inférieur à i${}_{\mu }$. Soient $F^{*}$ le dual de $F$, S$\left(F\right)$ l’algèbre symétrique de $F$, ${ℐ}_{\mu }$ l’idéal de ${𝒪}_X{\otimes }_{ℂ}\mathrm{S}\left(F\right)$ engendré par...

For any pair E,F of pseudotopological vector spaces, we endow the space L(E,F) of all continuous linear operators from E into F with a pseudotopology such that, if G is a pseudotopological space, then the mapping L(E,F) × L(F,G) ∋ (f,g) → gf ∈ L(E,G) is continuous. We use this pseudotopology to establish a result about differentiability of certain operator-valued functions related with strongly continuous one-parameter semigroups in Banach spaces, to characterize von Neumann algebras, and to establish...

We prove that a pure state on a $C^{*}$-algebras or a JB algebra is a unique extension of some pure state on a singly generated subalgebra if and only if its left kernel has a countable approximative unit. In particular, any pure state on a separable JB algebra is uniquely determined by some singly generated subalgebra. By contrast, only normal pure states on JBW algebras are determined by singly generated subalgebras, which provides a new characterization of normal pure states. As an application we contribute...