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##
Thierry Giordano - *Ergodic theory and dimension **G*-spaces

*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
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 *A*^{n} 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

where 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 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
*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.

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