Measures representing valuation maps on functions of bounded variation over idempotent semigroups

Abstract

Let S denote an idempotent semigroup, let W denote a Banach space. The space BV ( S, W), which is the space of functions of bounded variation from S into W, is considered. It is shown that if f is in BV ( S, W) and if W ∗∗ contains no copy of l ∞ then the value of f at every point is ∫ Γ γ( s) dμ f ( γ), where Γ is the structure space of S and μ f is an appropriate W valued measure. The hypothesis that W ∗∗ has no copy of l ∞ is then dropped and necessary and sufficient conditions are given for μ f to still have values in W. An application is made to Lipschitz functions and conditions are derived for μ ff to be a Gelfand or a Pettis indefinite integral. Another application is made to product measures.