  In this paper we extend a result of Semrl stating that every 2-local
automorphism of the full operator algebra on a separable infinite dimensional
Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we
obtain the same conclusion for the much larger class of Banach spaces with
Schauder bases. The proof rests on an analogous statement concerning the
2-local automorphisms of matrix algebras of which statement we present a short
proof. The need to get such a proof was formulated in Semrl's paper.
