Abstract
The aim of this paper is to study some of the relationships between groups of homeomorphisms on one side, and A.F C*-algebra, unitary commutative ring on the other side.
Let G be a countable group of homeomorphisms of a locally compact second countable topological space E. The class of an orbit O is the union of all orbits O' having the same closure as O. We denote by X the quasi-orbits space (i.e the space of orbits classes). If every decreasing sequence of saturated closed subsets of E is finite, then X is homeomorphic to the prime spectrum of a unitary commutative ring equipped with the Zariski topology and E is the closure of the union of a finitely many orbits.
Let E be the line IR such that every element of G is an increasing homeomorphism and let X-0 be the union of all open subsets of X homeomorphic to IR or S-1. The space X - X-0 is always homeomorphic to the primitive spectrum of an A.F C*-algebra equipped with the Jacobson topology and if G has a minimal set, then it is homeomorphic to the prime spectrum of a unitary commutative ring equipped with the Zariski topology if and only if every totally ordered family of orbits has a greatest lower bound.
We give an example of a diffeomorphism of the unit 2-sphere S-2 such that the above result fails.