A Lefschetz fixed-point formula for certain orbifold C*-algebras

Using Poincar'e duality in K-theory, we state and prove a Lefschetz fixed point formula for endomorphisms of cross product C*-algebras $C_0(X)\cross G$ coming from covariant pairs. Here $G$ is assumed countable, $X$ a manifold, and $X\cross G$ cocompact and proper. The formula in question expresses the graded trace of the map on rationalized K-theory of $C_0(X)\cross G$ induced by the endomorphism, \emph{i.e.} the Lefschetz number, in terms of fixed orbits and representation-theoretic data connected with certain isotropy subgroups of the isotropy group at that point. The technique is to use noncommutative Poinca'e duality and the formal Lefschetz lemma of the second author.

💡 Research Summary

The paper establishes a Lefschetz fixed‑point formula for a broad class of non‑commutative C*‑algebras that arise as crossed products C₀(X)⋊G, where G is a countable discrete group acting properly and cocompactly on a smooth manifold X. The authors work under the hypothesis that the action of G on X is proper (so that the isotropy groups Gₓ are finite) and that the quotient space X/G is compact. Under these conditions the crossed‑product algebra behaves like the C*‑algebra of an orbifold, and it admits a Poincaré duality in K‑theory: there is a natural isomorphism
K⁎(C₀(X)⋊G) ≅ K⁎(X) ⊗ K⁎(C*(G)),
where K⁎(C*(G)) is identified with the representation ring of G.

The main object of study is a covariant pair (φ,U) consisting of a continuous map φ:X→X and a unitary representation U:G→U(H) such that U(g) ∘ φ = φ ∘ U(g) for every g∈G. This pair determines a ‑endomorphism
α : C₀(X)⋊G → C₀(X)⋊G,
by the usual formula α(f δ_g) = (f∘φ) U(g) δ_g. On K‑theory α induces a graded homomorphism α_; its graded trace (the Lefschetz number) is defined as
L(α) = Tr(α_|K₀) − Tr(α_|K₁).

To compute L(α) the authors adapt the “formal Lefschetz lemma” of the second author, which in the commutative setting relates the trace of a K‑theory map to a sum over fixed points of the underlying geometric map. The non‑commutative version requires careful handling of the isotropy groups. For each fixed point x∈Fix(φ) the isotropy subgroup
Gₓ = {g∈G | g·x = x}
is finite. The representation U restricted to Gₓ decomposes into irreducibles; the authors single out a distinguished virtual representation Vₓ that captures the contribution of the local action of U(g) and the differential of φ at x. The contribution of the orbit of x to the Lefschetz number is then
(1/|Gₓ|)·Tr_{Vₓ}(U(gₓ)),
where gₓ is a representative element of the conjugacy class in Gₓ corresponding to the local linearization of φ. Summing over all G‑orbits of fixed points yields the main formula:

Theorem (Lefschetz formula for C₀(X)⋊G).
L(α) = ∑_{