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/