Indagationes Mathematicae-New Series
Invertibility in Banach algebras generated by two idempotents can be checked with the help of a theorem by Roch, Silbermann, Gohberg, and Krupnik. This theorem cannot be used to study generalized invertibility. The present paper is devoted to group invertibility in two types of finite-dimensional algebras which are generated by two idempotents, algebras generated by two tightly coupled idempotents on the one hand and algebras of dimension at most four on the other. As a side product, the paper gives the classification of all at most four-dimensional algebras which are generated by two idempotents. (c) 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
Böttcher, A., & Spitkovsky, I. M. (2012). Group inversion in certain finite-dimensional algebras generated by two idempotents. Indagationes Mathematicae, 23(4), 715-732.