Derivations and automorphisms of operator algebras

Abstract

The theorem that each derivation of aC*-algebra\(\mathfrak{A}\) extends to an inner derivation of the weak-operator closure ϕ(\(\mathfrak{A}\))− of\(\mathfrak{A}\) in each faithful representation ϕ of\(\mathfrak{A}\) is proved in sketch and used to study the automorphism group of\(\mathfrak{A}\) in its norm topology. It is proved that the connected component of the identity i in this group contains the open ball ℬ of radius 2 with centerl and that each automorphism in ℬ extends to an inner automorphism of ϕ(\(\mathfrak{A}\))−.