A Dichotomy for the Mackey Borel Structure

We prove that the equivalence of pure states of a separable C*-algebra is either smooth or it continuously reduces [0, 1]N/l2 and it therefore cannot be classified by countable structures. The latter was independently proved by Kerr–Li–Pichot by using different methods. We also give some remarks on a 1967 problem of Dixmier. If E and F are Borel equivalence relations on Polish spaces X and Y , respectively, then we say that E is Borel reducible to F (in symbols, E ≤B F ) if there is a Borel-measurable map f : X → Y such that for all x and y in X we have xEy if and only if f(x)Ff(y). A Borel equivalence relation E is smooth if it is Borel-reducible to the equality relation on some Polish space. Recall that E0 is the equivalence relation on 2 N defined by xE0y if and only if x(n) = y(n) for all but finitely many n. The Glimm–Effros dichotomy ([8]) states that a Borel equivalence relation E is either smooth or E0 ≤B E. One of the themes of the abstract classification theory is measuring relative complexity of classification problems from mathematics (see e.g., [12]). One can formalize the notion of ‘effectively classifiable by countable structures’ in terms of the relation ≤B and a natural Polish space of structures based on N in a natural way. In [10] Hjorth introduced the notion of turbulence for orbit equivalence relations and proved that an orbit equivalence relation given by a turbulent action cannot be effectively classified by countable structures. The idea that there should be a small set B of Borel equivalence relations not classifiable by countable structures such that for every Borel equivalence relation E not classifiable by countable structures there is F ∈ B such that F ≤B E was put forward in [11] and, in a revised form, in [4]. In this note we prove a dichotomy for a class of Borel equivalence relations corresponding to the spectra of C*-algebras by showing that one of the standard turbulent orbit equivalence relations, [0, 1]/l2, is Borel-reducible to every non-smooth spectrum. Date : August 25, 2009. 1991 Mathematics Subject Classification. 03E15, 46L30, 22D25.