Continuity of multiplication in projective limit algebras and applications to amenability

Abstract We study Köthe PDF-algebras. Using two (different yet natural) definitions of multiplication we obtain a wide class of natural algebras with either discontinuous or continuous multiplication. In this last case, we are able to fully characterize amenable Köthe PDF-algebras in terms of the defining Köthe matrix. This characterization shows an interesting and unexpected relation between algebraic and topological structures of amenable Köthe PDF-algebras.