SemigroupForumVol. 17 (1979) 261-265 RESEARCH ARTICLE THE MAAK COMPACTIFICATION OF A DENSE SUBSEMIGROUP H. D. Junghenn Communicated by D. E. Ramirez INTRODUCTION. Let S be a dense subsemi6roup of a topological semi- group T. In this note we investigate conditions under which the Maak compactlflcatlons of S and T coincide. The analog of this problem for the almost periodic and weakly almost periodic cases has been investigated by several authors. Hildebrant and Lawson proved that if S is an ideal of T and the almost periodic (resp. weakly almost periodic) compectiflcatlon of S is weakly reductlve, then S and T have the same almost periodic (resp. weakly almost periodic) compaotifications [~J . Milnee showed that the same conclusions hold if T is a topological group [71 • ( In [4J S and T were assumed to be only semltopological.) Our main result is the followings If S is a dense subsemlgroup of a topological semi- group T and if (tT~Tt)~S @ ~ V tsT, then S and T have the same Maak compactiflcations. Examples are given which show that the hypo- theses are satisfied for a large class of semigroups S and T. PRELIMINARIES Let S be a topological semlgroup and C(S) the C*-algebra of bounded continuous complex-valued functions on S. The space SAP(S) of ~ perl~¢ functions on S is by definition the closed linear span in C(S) of all coefficients of finite dimensional contin- uous unitary representations of S. A FreAk (or ~tron~lv almost periodic) ~om~ct!~ication of S is a pair (S,m), where S is a compact topolo6X- cal group and mrs-* S is a continuous homomorphism with dense range such that m*C(S) = SAP(S). Here m*s C(S)-+ C(S) is the dual mapping f "~ fore. Maak compactificatlons of S are unique up to isomorphism, this 261 0037-1912/79/0017-0261 $01.00 © 1979 Springer-VerlagNew York Inc.