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Lefschetz Numbers for $C^*$-Algebras

  Published:2010-08-03
 Printed: Mar 2011
  • Heath Emerson,
    University of Victoria, Victoria, BC
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Abstract

Using Poincar\'e duality, we obtain a formula of Lefschetz type that computes the Lefschetz number of an endomorphism of a separable nuclear $C^*$-algebra satisfying Poincar\'e duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on $\textrm{K}$-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.
MSC Classifications: 19K35, 46L80 show english descriptions Kasparov theory ($KK$-theory) [See also 58J22]
$K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
19K35 - Kasparov theory ($KK$-theory) [See also 58J22]
46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
 

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