A Non-commutative *-algebra of Borel Functions

Abstract

To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.