A not necessarily continuous, linear or multiplicative function £c from an algebra A into itself is called a local automorphism if £c agrees with an automorphism of A at each point in $A$. In this paper, we study the question when a local automorphism of a semisimple Banach algebra, is a Jordan isomorphism. Also a algebra is not necessary unital, but be implicitly assumed to be associative.

Let A be an algebra. A mapping : A ! A is called a 2-local automorphism if for every a, b in A there is an automorphism ab : A ! A, depending on a and b, such that ab(a) = (a) and ab(b) = (b). Here no linearity, surjectivity or continuity of is assumed. In this thesis we extend a result of Lajos Moln´ar stating that every 2-local automorphism of an operator algebra on a Banach space with a Schauder basis is an automorphism. We obtain the same conclusion for operator algebras on Fr´echet spaces with Schauder bases.