Isomorphisms in K-theory from isomorphisms in groupoid homology theories

We prove that for torsion-free amenable ample groupoids, an isomorphism in groupoid homology induced by an étale correspondence yields an isomorphism in the K-theory of the associated ${\mathrm{C}}^{\ast }$-algebras. We apply this to extend X. Li’s K-theory formula for left regular inverse semigroup ${\mathrm{C}}^{\ast }$-algebras. These results are obtained by developing the functoriality of the ABC spectral sequence.