On the Inversion Invariance of Invariant Means

An invariant mean on a group G is a normalized, positive, translation invariant linear functional defined on the space of all bounded complex valued functions on G. Some groups possess an invariant mean (or are said to be amenable), while others do not. In particular, all abelian groups are amenable [2, §17.5]. An invariant mean on a group need not be inversion invariant. It is quite easy to show, however, that if a group is amenable, then there exists an invariant mean that is also inversion invariant [2, §17.19a]. The purpose of this note is to prove that on a nontorsion abelian group, there must exist invariant means that are not inversion invariant. More specifically it is shown that no "elementary" (or extremal) invariant mean on a nontorsion abelian group is inversion invariant. For a finite abelian group, there exists a unique invariant mean which must be inversion invariant. For an infinite torsion abelian group, it is not clear just what the situation is. (See Remark 4.) While this note concerns itself only with the question of the inversion invariance of invariant means defined on the space of all bounded functions on an abelian group, this question is perhaps of even more interest in the more general context of invariant means defined on some "suitable" subspace of functions on an arbitrary (perhaps topological) group. The known results seem to indicate that this question may be related to the question of uniqueness of invariant means. Let G be an abelian group and let P(G) denote the complex Banach space of bounded complex valued functions on G in which ||/|| = sup {| fix) | | x G G}. An element PGP (G) * (the con j ugate space of BiG)) is said to be an invariant mean on G if (1) ||p|| = 1 ; (2) P/^ 0 for /^0; and (3) P/x = P/for each xEG and/GP(G), where/„(y) =/(xy) for y EG.1 An abelian group always has an invariant mean; for this result and further information about invariant means consult [2, §17]. For each function/GP(G), let/* denote the function defined on G such that /*(x) =/(x_1) for xGG. Similarly, for each functional PGP(G)*, let L* denote the functional defined on BiG) such that L*if) =P(/*) for each fEBiG). A functional PGP(G)* is said to be inversion invariant if L = L*. If L is an invariant mean on G, then the