## 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.

## Seminar (Operator Algebras)

Álvaro Sánchez (Universitat Autònoma de Barcelona) delivered the talk:

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

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 inK0(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.

## Seminar (Operator Algebras)

Eduard Vilalta (Universitat Autònoma de Barcelona) delivered the talk:

The real rank of uniform Roe algebras II

Abstract:

The aim of this 2-session seminar is to introduce the relation that has recently been found between the asymptotic dimension of a bounded geometry metric space X and the real rank of its associated uniform Roe algebra C*u(X) [1].
During the first session, I will give the definitions and results that will be needed for the second part. These include the real and stable rank of a C*-algebra[2], the asymptotic dimension of both a topological space and a group[3], and the uniform Roe algebra of a bounded geometry metric space[4].
In the second session, I will follow [1] to prove that, given a bounded geometry metric space X, the real rank of C*u(X) is 0 whenever the asymptotic dimension of X is 0. I will also explain the involvement of the first Chern class in the computation of the k0-group of C*u(Z^2), which is used in [1] to prove that the real rank of this algebra is non-zero.

[1] K. Li and R. Willet. “Low Dimensional Properties of Uniform Roe Algebras”. Journal of the London Mathematical Society, 97:98–124, 2018.
[2] L.G. Brown and G.K. Pedersen. “C*-Algebras of Real Rank Zero”. Journal of Functional Analysis, 99:131–149, 1991.
[3] G. Bell and A. Dranishnikov. “Asymptotic dimension”. Topology and its Applications, (155):1265–1296, 2008.
[4] N.P. Brown and N.Ozawa. “C*-Algebras and Finite-Dimensional Approximations”, volume 88 of Graduate Studies in Mathematics. American Mathematical Society, 2008.