Without the "scarcity lemma", two kinds of "rank one elements" are identified in semisimple Banach algebras.
@article{bwmeta1.element.bwnjournal-article-smv117i1p73bwm, author = {Robin Harte}, title = {On rank one elements}, journal = {Studia Mathematica}, volume = {113}, year = {1995}, pages = {73-77}, zbl = {0837.46036}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv117i1p73bwm} }
Harte, Robin. On rank one elements. Studia Mathematica, Tome 113 (1995) pp. 73-77. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv117i1p73bwm/
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