This paper investigates the $\mathrm{ K}$-theory of twisted groupoid $\mathrm{ C}^*$-algebras. It is shown that a homotopy of twists on an ample groupoid satisfying the Baum-Connes conjecture with coefficients gives rise to an isomorphism between the $\mathrm{ K}$-theory groups of the respective twisted groupoid $\mathrm{ C}^*$-algebras. The results are also interpreted in an inverse semigroup setting and applied to generalized Renault-Deaconu groupoids and $P$-graph algebras.
Categories: math.OA math.KT