Resumen de Local algebras and the largest spectrum finite ideal

Omar Jaa, Antonio Fernández López

• M. R. F. Smyth proved in [9, Theorem 3.2] that the socle of a semiprimitive Banach complex algebra coincides with the largest algebraic ideal. Later M. Benslimane, A. Kaidi and O. Jaa showed [3] the equality between the socle and the largest spectrum finite ideal in semiprimitive alternative Banach complex algebras. In fact, they showed that every spectrum finite one-sided ideal of a semiprimitive alternative Banach complex algebra is contained in the socle. In this note a new proof is given of this last result by using the notion of local algebra attached to an element of an (associative, alternative or Jordan) algebra. Only the associative case will be considered here since there is no essential difference between the associative and the alternative cases.