Continuous homomorphisms of Arens-Michael algebras

Abstract

It is shown that every continuous homomorphism of Arens-Michael algebras can be obtained as the limit of a morphism of certain projective systems consisting of Fr\'{e}chet algebras. Based on this we prove that a complemented subalgebra of an uncountable product of Fr\'{e}chet algebras is topologically isomorphic to the product of Fr\'{e}chet algebras. These results are used to characterize injective objects of the category of locally convex topological vector spaces. Dually, it is shown that a complemented subspace of an uncountable direct sum of Banach spaces is topologically isomorphic to the direct sum of ({\bf LB})-spaces. This result is used to characterize projective objects of the above category.