Invertibility preserving maps preserve idempotents (Q1590914)

Let \({\mathcal B}\) be a unital norm-closed operator algebra on some complex Banach space such that \({\mathcal B}\) contains all of the finite-rank operators. The main result of the paper under review is that, if \({\mathcal A}\) is a unital Banach algebra and \(\varphi:{\mathcal A}\to{\mathcal B}\) is a unital surjective linear mapping which takes every invertible element of \({\mathcal A}\) into an invertible element of \({\mathcal B}\), then \(\varphi\) also takes every idempotent element of \({\mathcal A}\) into an idempotent from \({\mathcal B}\). The results of this type are related to a question of \textit{I. Kaplansky} [``Algebraic and analytic aspects of operator algebras'', Conf. Board Math. Sci. Regional Conf. Ser. Math. No. 1. Providence, R.I.: (1970; Zbl 0217.44902)] asking for sufficient conditions such that a unital surjective linear invertibility preserving mapping be a Jordan homomorphism. As a consequence of the above cited main theorem, one proves in the paper under review that, if \({\mathcal A}\) is a von Neumann algebra, then every unital surjective linear invertibility preserving mapping \(\varphi:{\mathcal A}\to {\mathcal B}\) is a Jordan homomorphism, thus extending a result of \textit{A. R. Sourour} [Trans. Am. Math. Soc. 348, No. 1, 13-30 (1996; Zbl 0843.47023)].