Topologically stationary locally compact groups and amenability

Abstract

A semigroup S is called right stationary if for each/E m(S), the weak* closed convex hull of right translates off in m(S) = lx(S)* contains a constant function.In [18], T. Mitchell proved that a semigroup S is right stationary iff m(S) has a left invariant mean.In this case, the set of values p(f) where p runs over all left invariant means on m(S) coincides with the set of constants in the weak* closed convex hull of right translates off.The main purpose of this paper is to prove a topological analogue (which is also ,a generalisation) of this theorem for locally compact groups.It turns out that our analogue of Mitchell's theorem is a consequence of a more general "Localisation Theorem" on invariant means which we are going to present in §5.An application of this " Localisation Theorem" and its consequences is given in §6 while the rest of the paper is devoted to proving a theorem on topological left almost convergence as well as some further generalisations of discrete theory.2. Notations.For general terms in harmonic analysis, we shall follow Hewitt and Ross [14] (unless otherwise stated explicitly).Let G be a locally compact group with a fixed left Haar measure A and modular function A. Symbols like J" • • ■ dx or J" • • • dy will always denote integration with respect to A. Let BM(G) be the algebra (pointwise operations) of all real valued bounded A-measurable functions on G with supremum norm || ||u and N the closed ideal of all locally null functions in BM(G) ([14, Definition 11.26]).Let LX(G)=BM(G)/N be the quotient Banach algebra with quotient norm || || " (i.e.essential supremum norm).A linear functional m in Loe(G)* is called a mean if ess inffSm(f) ^ess sup/ for any feLx (G) (see [12, §2.1]).A mean m on L^G) is topological left (right) invariant if for every feLoe(G), <p eP(G) = {<p eLJfi) : <p^0, ||9>|i = l}, we have mHere/~ is the function defined by f~(x) =f(x~1), xeG(2).m is topological invariant if it is both topological left and topological right invariant.If X is a linear subspace of £,«,((7) containing the constants, X is said to be topological left (right) invariant if P(G) * X c X (X*P(G)~ c X).