Lefschetz numbers for C*-algebras

TL;DR

This paper develops a Lefschetz number formula for endomorphisms of certain C*-algebras using Poincare duality, and applies it to Cuntz-Krieger algebras to relate algebraic invariants to permutation dynamics.

Contribution

It introduces a Lefschetz number formula for C*-algebra endomorphisms satisfying Poincare duality and the Kunneth theorem, with explicit application to Cuntz-Krieger algebras.

Findings

Derived a polynomial formula for Lefschetz numbers based on matrix A

Connected permutation dynamics to Lefschetz number behavior

Provided a new invariant for endomorphisms of Cuntz-Krieger algebras

Abstract

Using Poincare duality, we formulate a formula of Lefschetz type which computes the Lefschetz number of an endomorphism of a separable, nuclear C*-algebra satisfying Poincare duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on K-theory tensored with the complex numbers, as in the classical case.) We then consider 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.