Let 𝓐 be a Banach algebra without nonzero finite dimensional ideals. Then every compact semiderivation on 𝓐 is a quasinilpotent operator mapping 𝓐 into its radical.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc67-0-10,
author = {Matej Bre\v sar and Yuri Turovskii},
title = {A note on compact semiderivations},
journal = {Banach Center Publications},
volume = {68},
year = {2005},
pages = {133-137},
zbl = {1067.47046},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc67-0-10}
}
Matej Brešar; Yuri Turovskii. A note on compact semiderivations. Banach Center Publications, Tome 68 (2005) pp. 133-137. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc67-0-10/