Let A be a semisimple Banach algebra with a linear automorphism σ and let δ: I → A be a σ-derivation, where I is an ideal of A. Then Φ(δ)(I ∩ σ(I)) = 0, where Φ(δ) is the separating space of δ. As a consequence, if I is an essential ideal then the σ-derivation δ is closable. In a prime C*-algebra, we show that every σ-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 σ-derivation expansion formula on zero products.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm190-2-7,
author = {Tsiu-Kwen Lee and Cheng-Kai Liu},
title = {Partially defined $\sigma$-derivations on semisimple Banach algebras},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {193-202},
zbl = {1183.46050},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm190-2-7}
}
Tsiu-Kwen Lee; Cheng-Kai Liu. Partially defined σ-derivations on semisimple Banach algebras. Studia Mathematica, Tome 192 (2009) pp. 193-202. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm190-2-7/