Invariant linear functionals

Abstract

Let B be a Banach space and let ℒ(B) denote the space of all bounded inear operators from B to B, which is a Banach algebra under composition of operators as multiplication. By a semigroup of operators G on B, we mean a norm bounded subser G of ℒ (B) which is a subsemigroup in the multiplicative structure of ℒ(B). The purpose of this paper is to study the existence of nonzero continuous linear functionals on B invariant under G, that is given B and G, does there exist μ∈B*, with μ ≠ 0, such that μ(Sx) = μ(x) for all x∈B and S∈G. This question is an attempt to generalize the familiar concepts of invariant means and amenability of semigroups. If H is any semigroup and m(H) is the Banach space of all bounded real valued functions on H with supremum norm, then a mean is a positive normalized continuous linear functional on m(H). A mean is called (left) [right] invariant if it is invariant under (left) [right] translations and H is called (left)[right] amenable if there exists such a mean; e.g., F is a left invariant mean if F∈m(H)* is such that (i) ‖F‖ = 1, (ii) F(x)≧0 if x(g)≧0 for all g∈H, (iii) F(xg) = F(x) for all x∈m(H) and g∈G where xg(h) = x(gh) for all h∈H. Amenability of semigroups has been studied extensively in recent years, for example see Day [2] or Hewitt and Ross [6] for an introduction, and Day [3] for a comprehensive survey.