TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 152, December 1970 FUNCTIONAL ANALYTIC PROPERTIES OF TOPOLOGICAL SEMIGROUPS AND TV-EXTREME AMENABILITYC) BY ANTHONY TO-MING LAU Abstract. Let S be a topological semigroup, LUC (5) be the space of left uniformly continuous functions on S, and A(5) be the set of multiplicative means on LUC (5). If (*) LUC (5) has a left invariant mean in the convex hull of A(5), we associate with S a unique finite group G such that for any maximal proper closed left translation invariant ideal / in LUC (5), there exists a linear isometry mapping LUC (C)// one-one onto the set of bounded real functions on G. We also generalise some recent results of T. Mitchell and E. Granirer. In particular, we show that 5 satisfies (*) iff whenever S is a jointly continuous action on a compact hausdorff space X, there exists a nonempty finite subset F of A"such that sF= FTor all s e S. Furthermore, a discrete semigroup S satisfies (*) iff whenever {Ts; s e S} is an antirepresentation of S as linear maps from a norm linear space X into X with II7*s||á 1 for all J65, there exists a finite subset aç5 such that the distance (induced by the norm) of x from Kx = linear span of {x — T,x; x e X, s e S} in X coincides with distance of O(o, *)={(l/|o-|) 2«e„ Tat(x); t eS} from 0 for all xe X. 1. Preliminaries and some notations. Let S be a topological semigroup (i.e. a set with an associative multiplication and a hausdorff topology such that for each ae S, the mappings j -> as and s^* s a, s e S, are continuous from S into S) and X a hausdorff topological space. An action of S on A' is a separately continuous mapping Sx X^> X(i.e. continuous in each one of the variables when the other one of the variables is kept fixed) denoted by (s, x)->s-x, such that s(t-x) = (s-t)x for all s, t e S and xe X. An action on X is jointly continuous if the mapping Sx X-*- AT is continuous when Sx X has the product topology. Let S be a topological semigroup which acts on a hausdorff topological space X, f be a bounded real function on X, s/(x)=/(sx) for all seS, xeX and 11/11 =supxeX \f(x)\ ;/is called S-uniformly continuous if/is continuous, and when- ever sa->s0, sa, s0 e S, then lima \\Saf—Sof\\=0. We shall denote by m(X) = the space of bounded real functions on X, - Received by the editors January 29, 1970 and, in revised form, February 3, 1970. AMS 1968subject classifications.Primary 4696; Secondary 2875, 2240, 5485. Key words and phrases. Topological semigroups, n-extremely amenable, amenable semi- group, n-extremely amenable semigroup, uniformly continuous functions, jointly continuous actions, multiplicative means, invariant means, maximal translation invariant closed ideal, finite intersection property, right ideal, group homomorphisms, locally compact groups, fixed points, point measure. (*) A portion of the results in this paper is contained in the doctoral thesis of the author written under the direction of Professor E. E. Granirer at the University of British Columbia. The author is most indebted to Professor Granirer for his suggestions and encouragement. Copyright © 1970, American Mathematical Society 431 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use