Abstract. We bound the Borel cardinality of the isomorphism relation for nuclear simple separable C * -algebras: It is turbulent, yet Borel reducible to the action of the automorphism group of the Cuntz algebra O2 on its closed subsets. The same bounds are obtained for affine homeomorphism of metrizable Choquet simplexes. As a by-product we recover a result of Kechris and Solecki, namely, that homeomorphism of compacta in the Hilbert cube is Borel reducible to a Polish group action. These results depend intimately on the classification theory of nuclear simple C * -algebras by K-theory and traces. Both of necessity and in order to lay the groundwork for further study on the Borel complexity of C * -algebras, we prove that many standard C * -algebra constructions and relations are Borel, and we prove Borel versions of Kirchberg's O2-stability and embedding theorems. We also find a C * -algebraic witness for a Kσ hard equivalence relation.

We prove that if V = L then there is a Π 1 1 maximal orthogonal (i.e. mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known Theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.2000 Mathematics Subject Classification. 03E15.

Correspondence to be sent to: asgert@math.ku.dk We establish the Borel computability of various C * -algebra invariants, including the Elliott invariant and the Cuntz semigroup. As applications we deduce that AF algebras are classifiable by countable structures, and that a conjecture of Winter and the second author for nuclear separable simple C * -algebras cannot be disproved by appealing to known standard Borel structures on these algebras.

Abstract. We prove that the isomorphism relation for separable C * -algebras, the relations of complete and n-isometry for operator spaces, and the relations of unital n-order isomorphisms of operator systems, are Borel reducible to the orbit equivalence relation of a Polish group action on a standard Borel space.

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver ([19]) and Harrington-Kechris-Louveau ([5]) show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on P(ω), the power set of ω, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on P(ω). In this article we examine the effective content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene's O as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington (see [6]) establishing for any recursive ordinal α the existence of Π 0 1 singletons whose α-jumps are Turing incomparable. * The authors acknowledge the generous support of the FWF through project P 19375-N18 ( * ) There are effectively Borel sets A and B such that for no effectively Borel function f does one have f [A] ⊆ B or f [B] ⊆ A.