TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 329, Number I, January 1992 COMPACTIFICATIONS OF LOCALLY COMPACT GROUPS AND CLOSED SUBGROUPS A. T. LAU, P. MILNES AND J. S. PYM ABSTRACT. Let G be a locally compact group with closed normal subgroup N such that GIN is compact. In this paper, we construct various semigroup compactifications of G from compactifications of N of the same type. This enables us to obtain specific information about the structure of the compactifi· cation of G from the structure of the compactification of N . Our results seem to be interesting and new even when G is the additive group of real numbers and N is the integers. Applications and other examples are given. 1. INTRODUCTION A key technique in the study of locally compact groups has been to "induce" properties of a group from information about its closed subgroups. This has been dramatically successful in the theory of group representations (Mackey [22], Dixmier [10], Kirillov [20]). In a less exalted sphere, there are theories al- lowing the extension of functions in various classes from a subgroup to the whole group (de Leeuw and Glicksberg [9], Dixmier [10], Henrichs [17], Berglund et al. [4]). In this paper, we shall also obtain results of the latter kind; though they will not be new, they will appear as corollaries to a new theory. Our principal concern is with semigroup compactifications '7F.9I.9, .91.9, etc.) of a group G and we show how to construct these from the same compactifica- tions of a closed normal subgroup N provided that G / N is compact and that some further conditions are satisfied (which always are when G is commuta- tive). As corollaries, besides results about extension of functions, we also find information about the structure of compactifications of G. We shall indicate the scope of our theory by describing its application in a special case, when G = lR and N = Z. Let 1= [0, 1]; then 1+ Z = lR. (More details are given in 5.3.) We interpret this as saying that lR can be obtained from Z by attaching a copy of I between each pair of points n, n + 1 (n E Z). If we take a universal compactification of Z, say Z'YN.9' (the weakly almost periodic compactification of Z), then our theory says that lR'YN.9' can be obtained by adjoining I between each pair x, x + 1 (x E Z'YN.9'). The same holds for other compactifications. Minimal left ideals in, say, can be obtained Received by the editors March IS, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 43A60, 22D05. Key words and phrases. locally compact group, subgroup, flow, compactification, weakly almost periodic function, distal function, uniformly continuous function, euclidean motion group. This research was supported in part by NSERC grants A7679 and A7857. 97 © 1992 American Mathematical Society 0002-9947/92 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use