Note on Reflexivity and Invariant Means

Applying Šmulian Theorem, we show that if a Banach space X is not reflexive then the space of bounded functions from Z with values in X does not admit an invariant mean. Invariant means on amenable groups are an important tool in many parts of mathematics, especially in harmonic analysis (see [5, 6]). For basic properties of invariant means, we refer the reader to [5]. We would only like to mention that a large class of “reasonable” groups is amenable, including abelian, solvable and finite. Invariant means and their generalizations for vector-valued functions play also an important role in the stability of functional equations and selections of set-valued functions (see, for example, [8, 3, 4, 7, 1]). Thus it seems natural to ask what are possible limitations of the use of invariant means. We will show that invariant means are, in some sense, naturally restricted to reflexive Banach spaces. Let G be a group and X be a Banach space. The space of all bounded functions from G into X is denoted by B(G, X). We are now ready to quote from [3] the definition of the generalization of invariant mean for vector-valued functions. Definition 1. We say that a linear function m : B(G, X) → X is an invariant mean if the following conditions hold: (i) for every f ∈ B(G, X) and a ∈ G there is mx(f(a + x)) = mx(f(x + a)) = m(f), where the subscript x next to m implies that the mean is taken with respect to the variable x. 2000 Mathematics Subject Classification. 43A07.