LAUSR

We provide a study of Köthe sequence algebras. These are Fréchet sequence algebras which can be viewed as abstract analogoues of algebras of smooth or holomorphic functions. Of particular treatment are the following properties: unitality, m‐convexity, Q‐property and variants of amenability. These properties are then checked against the topological (DN) ‐(Ω) type conditions of Vogt–Zaharjuta. Description of characters and closed ideals in Köthe echelon algebras is provided. We show that these algebras are functionally continuous and we characterize completely which of them factor. Moreover, we prove that closed subalgebras of Montel m‐convex Köthe echelon algebras are Köthe echelon algebras as well and we give a version of Stone–Weiestrass theorem for these algebras. We also emphasize connections with the algebra of smooth operators.