A Lefschetz fixed-point formula for certain orbifold C-algebras

Using Poincaré duality in K-theory, we state and prove a Lefschetz fixed point formula for endomorphisms of crossed product C*-algebras C0(X) ⋊ G coming from covariant pairs. Here G is assumed countable, X a manifold, and X ⋊ G cocompact and proper. The formula in question describes the graded trace of the map induced by the automorphism on K-theory of C0(X) ⋊ G, i.e. the Lefschetz number, in terms of fixed orbits of the spatial map. Each fixed orbit contributes to the Lefschetz number by a formula involving twisted conjugacy classes of the corresponding isotropy group, and a secondary construction that associates, by way of index theory, a group character to any finite group action on a Euclidean space commuting with a given invertible matrix.