Topology Proceedings 28 No. 2 (2004), pp. 503-526

Groups of homeomorphisms and spectral topology

H. Hattab and E. Salhi

The aim of this paper is to study some of the relationships between groups of homeomorphisms on one side, and approximately finite-dimensional (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 R such that every element of G is an increasing homeomorphism and let X₀ be the union of all open subsets of X homeomorphic to R or S¹. The space X-X₀ 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 S² such that the above result fails.

Keywords: Groups of homeomorphisms, quasi-orbits space, A.F. C^*-algebra, unit ary commutative ring

Mathematics Subject Classification: 57R30 57S05

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Topology Proceedings, Volume 28, No. 2 (2004)
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