Invariant means on topological semigroups
Loren N. Argabright · Pacific Journal of Mathematics · 1966
This paper is concerned with the existence and structure of invariant means on the space C(S) of all (including unbounded) continuous real-valued functions on a topological semigroup S. The main result is that for realcompact semigroups every left invariant mean (if any exist) arises as an integral over a compact left invariant subset of S. The question of existence of noncompact group G such that C(G) admits a left invariant mean is also considered.If G is a realcompact (or discrete, or locally compact abelian) group, then C(G) admits a left invariant mean only if G is compact.By a topological semigroup we mean a semigroup S endowed with a Hausdorff topology for which the mapping (x, y)->xy of S x S into S is continuous.We denote by C(S) the space of all continuous real-valued functions on S (not necessarily bounded).If a e S and fe C(S), then a f and f a will denote those functions on S whose values at x e S are f(ax) and f(xa) respectively.Obviously a f and f a are elements of C(S).A left invariant mean on C(S) is a nonnegative linear functional M on C(S) such that that M(ί) == 1 and M{J) = M(f) for all a e S and fe C(S).Eight invariant means are defined similarly, replacing a f by / α .A functional M that is both a left invariant mean and a right invariant mean is called a two-sided invariant mean.The purpose of this to study the structure of invariant means on C(S).This problem differs from the usual problem of finding invariant means for semigroups (see reference [2]), which concerns invariant means on the space C*(S) of all bounded continuous real-valued functions on S. Of course S is pseudocompact, then C(S) = C*(S) and the two problems are the same.Our main result is to the effect that, for a large class of semigroups S, every left invariant mean (if any exist) arises in a particularly simple way; namely, as an integral over a compact left invariant subset of S. In the final section we consider the question of whether there exist noncompact groups G such that C(G) admits a left invariant mean.A negative answer is obtained for noncompact groups of various types.In this section we establish some terminology and preliminary facts needed for the statement and proof of the main theorem.