Seminar (Operator Algebras)

Álvaro Sánchez (Universitat Autònoma de Barcelona)

Natural embedding of H0 (G) into K0 (Cr ∗ (G)) for rank 3 Deaconu-Renault groupoids, and HK conjecture II

Abstract: Take an étale groupoid G such that G (0) is compact, metrizable and totally disconnected.
By definition of Cr ∗ (G), we can always consider the canonical inclusion

ι : C(G(0)) →Cr ∗(G), which induces an homomorphism in K-theory

K0 (ι) : K0(C(G (0))) → K0 (Cr ∗ (G)).
Now, since there are no non-unit elements in G (0), K0 (C(G (0) )) = C(G (0) , Z). For any U compact open bisection of G, u = χU is a partial isometry of Cr ∗ (G) such that uu∗ = χs(U), and u∗ u = χr(U ) , which means χs(U ) and χr(U ) belong to the same equivalence class in K0(Cr ∗ (G)).

Then the differential map δ1 : Cc (G, Z) → C(G (0) , Z) defining the homology groups verifies (K0 (ι) ◦ δ1 )(u) = 0 (since δ1 (u) = χs(U ) − χr(U ) ). From here we deduce that imδ1 ⊆ ker(K0 (ι)). Since every étale groupoid has a countable basis consisting in open bisections, it follows that there exists a canonical homomorphism Φ : H0 (G) → K0 (Cr ∗(G)) such that Φ([f ]) := K0 (ι)(f ). The question about when this map is injective is open, even for some simple groupoids. We prove this result for the Deaconu-Renault groupoids of
rank 3, i. e. the groupoids arising from N3 acting over a Cantor set by surjective local homeomorphisms. We use this to prove the HK-conjecture for this family of groupoids.

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