Isomorphism of commutative group algebras of closed -local algebraically compact groups

Let be an abelian group and let be a field of . It is shown via a universal algorithm that if the modified Direct-Factor Problem holds, then the -isomorphism for some group yields provided is a closed -group or a -local algebraically compact group. In particular, this is the case when is closed -primary of arbitrary power, or is -local algebraically compact with cardinality at most and is in cardinality not exceeding . The last claim completely settles a question raised by W. May in Proc. Amer. Math. Soc. (1979) and partially extends our results published in Rend. Sem. Mat. Univ. Padova (1999) and Southeast Asian Bull. Math. (2001).