The Banach-Lie Group of Lie Automorphisms of an $H^*$-Algebra

Calderón Martín, Antonio J. ; Martín González, Candido

Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 623-631

Résumé

We study the Banach-Lie group $\operatorname{Aut}(A^-)$ of Lie automorphisms of a complex associative $H^*$-algebra. Also some consequences about its Lie algebra, the algebra of Lie derivations of $A$, are obtained. For a topologically simple $A$, in the infinite-dimensional case we have $\operatorname{Aut}(A^-)_0 = \operatorname{Aut}(A)$ implying $\operatorname{Der}(A) = \operatorname{Der}(A^-)$. In the finite dimensional case $\operatorname{Aut}(A^{-})_{0}$ is a direct product of $\operatorname{Aut}(A)$ and a certain subgroup of Lie derivations $\delta$ from $A$ to its center, annihilating commutators.