The authors consider unital algebras with the following property: for each , there exists with . Their main result states that such a -algebra (completely metrizable locally convex algebra) with an open group of invertible elements is finite-dimensional. Using this result, the authors show that a locally -algebra with the above property is an inverse limit of finite-dimensional algebras. Another result states that such an -algebra (completely metrizable algebra) is a finite product of division algebras of type .
Reviewer’s remark. It remains open whether such a division algebra must be finite-dimensional, i.e., equal to or .