Higher Point Derivations on Commutative Banach Algebras, II

This paper continues the study of continuity properties of higher point derivations on commutative Banach algebras. It is shown that there are uniform algebras and regular algebras which have totally discontinuous point derivations of infinite order. This shows that there can be no special automatic continuity results for higher point derivations either for uniform algebras or for regular algebras. It is also shown that there are Banach algebras which have homomorphisms onto the algebra of complex formal power series. An important element in the constructions is a topological version of the symmetric tensor algebra over a vector space. These algebras may be of independent interest; related objects have already been considered by other authors.