The stabilization theorem for proper groupoids

The stabilization theorem for $A$-Hilbert modules was established by G. G. Kasparov. The equivariant version, in which a locally compact group $H$ acts properly on a locally compact space $Y$, was proved by N. C. Phillips. This equivariant theorem involves the Hilbert $(H,C_{0}(Y))$-module $C_{0}(Y,L^{2}(H)^{\infty})$. It can naturally be interpreted in terms of a stabilization theorem for proper groupoids, and the paper establishes this theorem within the general proper groupoid context. The theorem has applications in equivariant KK-theory and groupoid index theory.