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Lefschetz Numbers for C*-Algebras

  • Heath Emerson (a1)

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 $\text{K}$ -theory tensored with $\mathbb{C}$ , 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.

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References

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Lefschetz Numbers for C*-Algebras

  • Heath Emerson (a1)

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