A note on -summable groups

We answer questions raised by P. Danchev in a recent paper in these Proceedings. It is shown that a -summable abelian -group is not determined by its socle, that is, two such groups can have isometric socles without being isomorphic. It is also demonstrated that -summability plays essentially no role in regard to the question of whether or not is totally projective, where denotes the group of normalized units of the group algebra with being a perfect field of characteristic .