Convergence of the CBHD series and associativity of the CBHD operation

Abstract

THE aim of this chapter is twofold. On the one hand, we aim to study the 4 convergence of the Dynkin series (Formula presented.) in various contexts. For instance, this series can be investigated in any nilpotent Lie algebra (over a field of characteristic zero) where it is actually a finite sum, or in any finite dimensional real or complex Lie algebra and, more generally, its convergence can be studied in any normed Banach-Lie algebra (over R or C). For example, the case of the normed Banach algebras (becoming normed Banach-Lie algebras if equipped with the associated commutator) will be extensively considered here.