PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 55, Number 2, March 1976 SEQUENTIAL CONVERGENCE TO INVARIANCE IN BC(G) ROBERT SINE Abstract. In this note it is shown that weak and strong convergence to invariance are equivalent for a sequence of probabilities acting on BC(G) of a noncompact locally compact group. This result was known for G = Z. For other generalizations of the bounded sequences on Z, say BUC(G), the result does not hold. 1. Introduction. Let G be a locally compact group (not necessarily abelian or even amenable). Our group will always be Hausdorff and this together with local compactness implies that G is a normal topological space [3, p. 76]. The sup-norm algebra of bounded continuous real valued functions on G will be denoted by BC{G). For any element y in G we denote the left translation operator byy on BC{G) by Tyf{x) = f{yx). A mean is a member, m, of the dual space BC{G)* with m{l) = 1 and / > 0 implying m{f) > 0. (Thus a mean can be regarded as a regular Borel normalized measure on the Stone-Cech compactification of G.) A mean m is left invariant if m{Tyf) = m{f) for all y in G and all / in BC{G). We will denote the (possibly empty) set of all left invariant means of A. As is suggested by G. G. Lorentz's theory of almost convergent sequences [5] we define £ = A± = {/in BC{G): m{f) = 0 for all m in A}. If BC{G) fails to be amenable then A = 0 and £ = BC{G). Let {mn) be a sequence of probabilities (regular Borel normalized mea- sures) on G. We say that {mn} acts on £ if {mn,f)—>0 for all / in £. Following M. M. Day [1] we say that {mn} converges weakly to invariance if (/— T)*mn—>0 [co*] for all y in G and converges strongly to invariance if ||(/ - Ty )*rtj„||—> 0 for all y in G.The norm here is the norm of the dual space of BC{G). It is clear that for measures on G this norm is the usual total vari- ation norm. 2. Results. Some of the results here hold for any translation invariant subspace on G but we obtain our main result only for BC{G). For this reason the entire discussion is in terms of this space. Theorem 1. £ = span (J {(/ - Ty)f: y in G and f in BC{G)). Proof. It is clear that any function of the form (/ - Ty)f is in £ and since £ is a norm closed subspace we have the inclusion of the hull in £. If the hull Received by the editors February 26, 1975. AMS (MOS) subject classifications (1970). Primary 43A07, 43A55, 46E15, 47A35, 60B10. Key words and phrases. Convergence to invariance, Banach limits, means on groups, almost convergence. © American Mathematical Society 1976 313 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use