AN EQUIVARIANT VERSION OF THE HAHN-BANACH THEOREM

Abstract

We prove an equivariant version of the Hahn-Banach Theorem, that allows simultaneous access to Day's [DAY 1] versions of the Hahn-Banach Theorem and the Krein-Rutman Extension Theorem [K&R 1] as well as related Theorems due to Wittstock [WIT 1] and Arveson [ARV 1]. We discuss an order-theoretical characterization of injective, unital C*-algebras. It can be used to get simple proofs of stability properties of injective W*-algebras. Our main application is the proof of a conjecture by Silverman [SIL 1]: Let (L, L+) be an ordered vector space with the least upper bound property. Suppose that S is a right amenable, discrete semigroup acting identically on L. If V is a real vector space with a representation of S as linear operators on V, and theta : V --> L is a sublinear map satisfying theta . sigma less-than-or-equal-to theta, sigma is-an-element of S, then there exists an S-equivariant, linear map theta : V --> L satisfying phi less-than-or-equal-to theta. In particular a discrete semigroup S is right amenable if and only if a S-equivariant Hahn-Banach principle for the real numbers is valid.