# Convergence of the CBHD Series and Associativity of the CBHD Operation

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2034)

## Abstract

THE aim of this chapter is twofold. On the one hand, we aim to study the 4 convergence of the Dynkin series
$$\begin{array}{*{20}c} {uv: = } & {\sum\limits_{j = 1}^\infty {\left( {\sum\limits_{n = 1}^j {\frac{{\left( { - 1} \right)^{n + 1} }}{n}\,\sum\limits_{\begin{array}{*{20}c} {(h_1,k_1 ), \cdots (h_n,k_n ) \ne (0,0)} \\ {h_1 + k_1 + \cdots + h_n + k_n = j} \\ \end{array}} \times \frac{{(ad\,u)^{h_1 } (ad\,\upsilon )^{k_1 } \cdots (ad\,u)^{h_n } (ad\,\upsilon )^{k_n - 1} (\upsilon )}}{{h_1 ! \cdots h_n !k_1 ! \cdots k_n !(\sum\nolimits_{i = 1}^n {(h_i + k_i )} )}}} } \right),} } \\ \end{array}$$
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.

## Keywords

Banach Algebra Associative Algebra Formal Power Series Magnus Expansion Maclaurin Expansion
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview 