Blog

Maria Stella Adamo (University of Rome “Tor Vergata”)

Cuntz-Pimsner algebras associated to C*-correspondences over commutative C*-algebras

Abstract: In this talk, structural properties of Cuntz-Pimsner algebras arising by full, minimal, non-periodic, and finitely generated C*-correspondences over commutative C*-algebras will be discussed. A broad class of examples is provided considering the continuous sections $Gamma(V,varphi)$ of a complex locally trivial vector bundle $V$ on a compact metric space $X$ twisted by a minimal homeomorphism $varphi: Xto X$. In this case, we identify a “large enough” C*-subalgebra that captures the fundamental properties of the containing Cuntz-Pimsner algebra. Lastly, we will examine conditions when these C*-algebras can be classified using the Elliott invariant. This is joint work in progress with Archey, Forough, Georgescu, Jeong, Strung, Viola.

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.

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.

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.

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

The real rank of uniform Roe algebras I

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.

Joan Claramunt (Universitat Autònoma de Barcelona) delivered the talk:

A correspondence between dynamical systems and separated graphs II

Abstract: In 1992 Herman, Putnam and Skau established (following the work of Versik) a bijective correspondence between essentially simple ordered Bratteli diagrams and essentially minimal dynamical systems. This correspondence enable the authors to study a particular subfamily of C*-crossed products (i.e. C(X) x Z given by a single homeomorphism f : X -> X; here X is the Cantor set). In these 2-session seminars I would like to present the work obtained so far in extending the above correspondence between dynamical systems (not necessarily minimal) and (a special class of) separated graph algebras. In the first session I will introduce the basic definitions, concepts and known results which will be used throughout the 2-session seminar. In the second session I will concentrate on presenting the work obtained so far, which is joint work in progress with P. Ara and M. S. Adamo.