Schurian Algebras and Spectral Additivity

A class of algebras is introduced that includes the unital Banach algebras over the complex numbers. Commutator results are proved for such algebras and used to establish spectral properties of certain elements of Banach algebras. 1. Introduction. In this note, we introduce a condition on an algebra motivated by the situation in which Schur's lemma [S1] is applicable. We say that algebras satisfying this condition are Schurian. For such algebras, we prove some (purely algebraic) results characterizing those elements associated with inner derivations having range in the radical. We note that, among others, each Banach algebra over the complex numbers C is Schurian. Our algebraic results are then applied, in conjunction with the theory of sub-harmonic functions — notably, Vesentini's result [V] on subharmonicity of the spectral radius (of holomorphic, Banach-algebra-valued functions from a domain in C) — to determine those elements in a Banach algebra that exhibit certain spectral properties (spectral additivity). These spectral-additivity results are themselves applied, in [H-K], to a topic initiated by Frobenius [F] (and contributed to by Schur [S2]), the study of mappings that preserve invertible elements.