Lefschetz numbers for C*-algebras

Abstract

Using Poincaré duality, we obtain a formula of Lefschetz type that computes the Lefschetz number of an endomorphism of a separable nuclear C*-algebra satisfying Poincaré duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on K-theory tensored with C, as in the classical case.) We then examine endomorphisms of Cuntz-Krieger algebras O A. An endomorphism has an invariant, which is a permutation of an infinite set, and the contracting and expanding behavior of this permutation describes the Lefschetz number of the endomorphism. Using this description, we derive a closed polynomial formula for the Lefschetz number depending on the matrix A and the presentation of the endomorphism. © Canadian Mathematical Society 2010.