Thierry Giordano - Ergodic theory and dimension G-spaces

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Thierry Giordano - Ergodic theory and dimension G-spaces

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 $A = \Bbb R G$ has a natural order structure given by the positive cone $A_+ = \{\sum r_g g ; r_g = 0$ a.e. and $r_g \ge 0\}$ and is endowed with an order-preserving action of G (by right multiplication). If $n \in \Bbb N$ , 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

$\begin{displaymath}H = \lim\limits_{\longrightarrow} A^{n(i)} \buildrel \phi_i\over \longrightarrow A^{n(i+1)} , \leqno(1) \end{displaymath}$

where $\phi_i$ is a positive A-module map (for the natural ordering on A).

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 $\gamma$ 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 $g \in G$ , $g\gamma \le N(g)\gamma$ (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 $(H,\gamma)$ which extends to L¹(X) and its dual.

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