Idempotent Measures on Compact Semigroups

Throughout this paper, 5 will be a compact Hausdorff topological semigroup, 5 will denote the convolution semigroup of normalized non-negative regular Borel measures on 5, and H will be the carrier of a measure p in 5. The space of continuous complex valued functions on 5 will be denoted by C(S), while Cr(S) will be the subspace of C(S) of real valued functions. Standard terminology and definitions used here may be found in the references numbered [2; S]. The main result of this paper (Theorem 3) is an intrinsic description (in terms of S, H, and C(S)) of the idempotents p in 5 whose carrier is H. Some of the conditions obtained in this theorem are somewhat artificial, and so are not entirely satisfactory. However, the main strength of the theorem is that part which says that the carrier H of each idempotent measure is a compact completely simple semigroup of 5. This result alone has proved quite useful (e.g., see [l]). Additional useful information contained in Theorem 3 is the statement that the idempotent measure, while not necessarily right invariant on H, does have some invariance properties when restricted to each minimal left ideal of H. Preliminary to these theorems we state and prove several lemmas needed in the sequel, the last of which gives a measure theoretic characterization of compact completely simple semigroups. In addition, some additional notation is needed. If gEC(S) and xES let gx(y) = g(yx), s0 that gxEC(S). If now pES, define g'(x) =p(gx); it is known [2, p. 53] that g'EC(S) as well. It is then clear that a mapping (for fixed p) of C(S) into C(S) is thus defined, and no ambiguity can result in the theorems in which this notation appears if it is understood that these functions g' are always defined in terms of a single p. Occasionally it will be expedient to employ the notation g", where by definition g" = (g')'; i.e., g"(x) =p[(g')x], with g'(t)=p(gl).