Partially defined $\sigma $-derivations on semisimple Banach algebras

Tsiu-Kwen Lee¹ ; Cheng-Kai Liu²

¹Department of Mathematics National Taiwan University Taipei 106, Taiwan
²Department of Mathematics National Changhua University of Education Changhua 500, Taiwan

Studia Mathematica, Tome 190 (2009) no. 2, pp. 193-202

Résumé

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.

Détail

DOI : 10.4064/sm190-2-7

Keywords: semisimple banach algebra linear automorphism sigma delta colon rightarrow sigma derivation where ideal nbsp phi delta cap sigma where phi delta separating space delta consequence essential ideal sigma derivation delta closable prime * algebra every sigma derivation defined nonzero ideal continuous finally linear map prime semisimple banach algebra nontrivial idempotents continuous satisfies sigma derivation expansion formula zero products