Partially defined $\sigma $-derivations on semisimple Banach algebras
Volume 190 / 2009
Studia Mathematica 190 (2009), 193-202
MSC: 46H40, 47B47, 46H15.
DOI: 10.4064/sm190-2-7
Abstract
Let $A$ be a semisimple Banach algebra with a linear automorphism ${\sigma}$ and let $\delta\colon I\rightarrow A$ be a ${\sigma}$-derivation, where $I$ is an ideal of $A$. Then $\Phi(\delta)(I\cap{\sigma}(I) )=0$, where $\Phi(\delta)$ is the separating space of $\delta$. As a consequence, if $I$ is an essential ideal then the ${\sigma}$-derivation $\delta$ is closable. In a prime $C^*$-algebra, we show that every $\sigma$-derivation defined on a nonzero ideal is continuous. Finally, any linear map on a prime semisimple Banach algebra with nontrivial idempotents is continuous if it satisfies the ${\sigma}$-derivation expansion formula on zero products.
