Free products of measured equivalence relations

Abstract

For given countable standard equivalence relations over the same unit space we construct their free product, which is a measured groupoid. Related topics such as measures and 2-cocyles on the free product (needed to deal with groupoid von Neumann algebras) are also discussed. r

Fullness, Connes’ $\chi $-groups, and ultra-products of amalgamated free products over Cartan subalgebras

“…To conclude this section, we would like to discuss a relation with work of Kosaki [12]. As was mentioned before, the triple A ⊇ D ⊆ B produces two countable nonsingular Borel equivalence relations R A , R B over a standard Borel probability space (X, µ).…”

Notes on Treeability and Costs for Discrete Groupoids in Operator Algebra Framework

“…If no alternating word in Γ 1 \Γ 3 , Γ 2 \Γ 3 with Γ 3 := Γ 1 ∩Γ 2 intersects with the unit space of strictly positive measure, i.e., Γ is the "free product with amalgamation Γ 1 ⋆ Γ 3 Γ 2 " (modulo null set), and Γ 3 is principal and hyperfinite, then the above formula immediately implies the formula C µ (Γ) = C µ (Γ 1 ) + C µ (Γ 2 ) − C µ (Γ 3 ) as long as when C µ (Γ 1 ) and C µ (Γ 2 ) are both finite. Here, we need the same task as in [21].…”

$L^2$-Betti numbers and costs in the framework of discrete groupoids

“…A rigorous (i.e., measurable) construction of free products with amalgamations was given in [14], but we do not need it here.…”