On continuity of the spectral radius function in Banach algebras

Summary This paper deals with the problem of the continuity of the spectral radius function in abstract Banach algebras. A new sufficient condition for the continuity of the function above at a point of a Banach algebra (which generalizes the already known ones and, in the particular case of the algebra of all linear and continuous operators on a separable Hilbert space, is also necessary for continuity of spectral radius) is given. Such a condition, in the algebra of all linear and continuous operators on a generic Banach space, is less restrictive than the already known ones, as several examplex show.