|THIERRY GIORDANO, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada|
|Ergodic theory and dimension G-spaces|
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 order-preserving action of G (by right multiplication). If , then An is a partially ordered vector space with the direct sum ordering and a G-space with the above G-action.
Definition. A G-dimension 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 matrix-valued 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 pseudo-norm on H and allows us to associate to H the real L1-space L1(X), the ``completion'' of H (as defined by Goodearl and Handelman). If for all , (N(g)depending on g), then G acts on L1(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 L1(X) and its dual.