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.
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Siegfried Echterhoff, Heath Emerson, Hyun Jeong Kim, A Lefschetz fixed-point formula for certain orbifold C*-algebras. J. Noncommut. Geom. 4 (2010), no. 1, pp. 125–155DOI 10.4171/JNCG/51