  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.

Operator Algebras