Continuité des dérivations et des épimorphismes dans certaines algèbres de Banach

Topologically simple Banach algebras with derivation

“…It follows that every derivation on A is continuous if C is the only commutative, topologically simple Banach algebra. The results from [8,11,9] all give, in one way or another, a necessary condition for / to be zero.…”

“…Partial results -all positive -are given, for instance, in [8], [11], and [9]. All these results utilise the connections of the automatic continuity question for derivations on commutative, semiprime Banach algebras with the perhaps deepest open problem in general Banach algebra theory, the closed ideal problem, which were first discovered by Cusack [5].…”

“…However, since the closed ideal problem has been open since the early days of Banach algebra theory, and no solution even seems to be in sight, we feel justified to proceed this way in view of [11,Theorem 6]: If all derivations on commutative, topologically simple Banach algebras are automatically continuous, then all derivations on commutative, semiprime Banach algebras are continuous. Certainly, the ideas contained in [8,11,9] do not help us any further. Instead, we shall borrow heavily from the first part of Thomas' proof of the Singer-Wermer conjecture [15], and show that, under certain additional conditions on D, a certain quotient of a localisation of A has a power series structure.…”