ABSTRACT
The Glimm-Effros dichotomy is undoubtedly the most important theoretical result that helped form the subject of invariant descriptive set theory. In the context of operator algebra Glimm and Effros have obtained the dichotomy for locally compact group actions and more generally Fσ orbit equivalence relations. Nowadays the terminology usually refers to the remarkable theorem for all Borel equivalence relations proved by Harrington, Kechris, and Louveau. The theorem and its proof were so influential that for a while the main activity of the field was to prove new dichotomy theorems. For orbit equivalence relations the Glimm-Effros dichotomy have been obtained by Solecki, Hjorth, Becker, and others. Another remarkable connection with other fields of mathematics is the study of hyperfinite equivalence relations. To this date there are still intriguing open problems around which exciting research is actively going on.
