Let V be a variety of not necessarily associative algebras, and A an inverse
limit of nilpotent algebras A_i\in V, such that some finitely generated
subalgebra S \subseteq A is dense in A under the inverse limit of the discrete
topologies on the A_i.
A sufficient condition on V is obtained for all algebra homomorphisms from A
to finite-dimensional algebras B to be continuous; in other words, for the
kernels of all such homomorphisms to be open ideals. This condition is
satisfied, in particular, if V is the variety of associative, Lie, or Jordan
algebras.
Examples are given showing the need for our hypotheses, and some open
questions are noted.Comment: Apologies; in submitting version 2, I didn't realize I had to delete
version 1; so the result was a mess. Here is the proper revised version. 23
pages, to appear, Ill.J.Math. Main changes in Aug.2010 revision: Re-formatted
to fit journal-sized page. New Section 8 added, on question of when
subalgebras of finite codimension must be open. Proofs of Lemma 5 and of
Lemma 6(iii) shortene
