Equivalence and stable isomorphism of groupoids, and diagonal-preserving   stable isomorphisms of graph C*-algebras and Leavitt path algebras
  We prove that ample groupoids with sigma-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matui's notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph C*-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras $L_Z(E_2)$ and $L_Z(E_{2-})$ are not stably *-isomorphic. 
Toke Meier Carlsen, Efren Ruiz and Aidan Sims
math.OA