
THIERRY GIORDANO, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada 
Ergodic theory and dimension Gspaces 
Let G be a discrete group. The real group algebra has a natural order structure given by the positive cone a.e. and and is endowed with an orderpreserving action of G (by right multiplication). If , then A^{n} is a partially ordered vector space with the direct sum ordering and a Gspace with the above Gaction.
Definition. A Gdimension space H is a partially ordered vector space with an action of G (as a group of order automorphisms) that can be obtained as a direct limit
Corresponding to the inductive limit in (1) is a matrixvalued random walk on G. The harmonic functions associated to this random walk are in a natural bijection with the states on H. A state corresponding to a bounded harmonic function is called bounded. It induces a pseudonorm on H and allows us to associate to H the real L^{1}space L^{1}(X), the ``completion'' of H (as defined by Goodearl and Handelman). If for all , (N(g)depending on g), then G acts on L^{1}(X).
D.E. Handelman and I have defined the notion of ergodicity and different generalizations of approximate transitivity for the action of G on which extends to L^{1}(X) and its dual.