A Selector for Equivalence Relations with G δ Orbits

Abstract

Assume is a Polish space and E is an open equivalence on X such that every equivalence class is a Gs set. We show that there is a Gs transversal for E. It follows that for any separable C*-algebra A, there is a Borel cross-section for the canonical map Iit(A) -> Prim(/4). Let Ibea Polish space, E an open equivalence relation on X. It is known [4, Corollary 2] that if all equivalence classes (orbits) are closed, then there is a Gs transversal for E. We will show that this result holds under the weaker hypothesis that each orbit is Gs. The existence of a Borel selector in the special case where each orbit is both Fg and Gs was established by Kallman and Mauldin [3]. Their result in turn extends the selector theorem in Effros [2]. Both the main theorem and Corollary 2 were conjectured in [3]. Remark on terminology. Given a space X and equivalence E, let m: X -» X/E be the canonical projection. A cross-section is a map s: X/E -» X such that tr ° s is the identity. A selector is a map /: X —> X which factors as a composition / = s ° it with s a cross-section. A selector is continuous (respectively Borel measurable) if and only if the associated cross-section is continuous in the quotient topology (resp. Borel measurable in the quotient Borel structure). A transversal is a subset of X which meets each orbit in a singleton. If / is a continuous (resp. 1-Borel measurable) selector, then Image(/) is a closed (resp. Gs) transversal. The converse does not hold in general. Our main theorem is slightly more general than the result promised in the first paragraph. Theorem 1. Let X be a Polish space, % a countable basis for the topology on X. Suppose E is an equivalence on X such that (i) For every O E %, the E-saturation of O is both Fa and Gs, (ii) Every Eorbit is Gs. Then there is a selector for E which is Borel measurable at level 1. It follows immediately that the associated transversal is Gs and the quotient Borel structure on X/ E is standard in the sense of Mackey [6]. Presented to the Society, January 5, 1978; received by the editors November 22, 1977 and, in revised form, January 26, 1978. AMS (MOS) subject classifications (1970). Primary 54H05, 54C65; Secondary 46L05, 54A10, 54B15.