On central traces and groups of symmetries of order unit Banach spaces

Abstract

A central trace on an order-unit Banach space A(K) is a centre-valued module homomorphism invariant under the group of symmetries of A(K).The concept of central traces has been crucial in the theory of types for convex sets established in (4), (5). In von Neumann algebras, they are precisely the canonical centre-valued traces and their existence hinges on a fundamental theorem (Dixmier's approximation process) in von Neumann algebras. On the other hand, the existence of central traces in finite dimensional spaces is an easy consequence of Ryll-Nardzewski's fixed point theorem (5).