A Lefschetz fixed-point formula for certain orbifold C*-algebras (Q2269733)

Summary: Using Poincaré duality in K-theory, we state and prove a Lefschetz fixed point formula for endomorphisms of crossed product \(C^*\)-algebras \(C_{0}(X) \rtimes G\) coming from covariant pairs. Here \(G\) is assumed countable, \(X\) a manifold, and \(X \rtimes G\) cocompact and proper. The formula in question describes the graded trace of the map induced by the automorphism on the K-theory of \(C_{0}(X) \rtimes G\), 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.