## Seminar (Operator Algebras)

Joachim Zacharias (University of Glasgow)

AF-embeddings and quotients of the Cantor set

Abstract: The classical Aleksandrov-Uryson Theorem says that every compact metric space X is a quotient of the Cantor set S, hence the C*-algebra C(X) of continuous functions on X embeds into C(S), an AF algebra, i.e. an inductive limit of finite dimensional C*-algebras. Thus every separable commutative C*-algebra is AF-embeddable. Whilst this cannot be true for arbitrary separable non-commutative C*-algebras such embeddings into AF-algebras have been established in many cases. We explore how the proof of the classical A-U-Theorem can be mimicked to obtain AF-embeddings and related results for classes of non-commutative C*-algebras.

## Seminar (Ring Theory)

Ferran Cedó (Universitat Autònoma de Barcelona)

Construcció de noves braces finites simples

Resum: Aquest és un treball conjunt amb l’Eric Jespers i el Jan Okninski. Donat un grup abelià finit A qualsevol, explicaré com construir braces simples finites amb grup multiplicatiu metabelià (és a dir, amb longitud derivada 2) tals que $A$ és isomorf a un subgrup del seu grup additiu. Abans d’aquest treball, cap de les braces simple finites conegudes contenia elements amb ordre additiu  4. En un treball anterior (junt amb David Bachiller, Eric Jespers i Jan Okninski), s’havien construït braces finites simples tals que el seu grup additiu contenia qualsevol grup abelià prefixat d’ordre senar, però el grup multiplicatiu d’aquestes braces era de longitud derivada 3.

## Seminar (Ring Theory)

Eric Jespers (Vrije Universiteit Brussel)

Associative structures associated to set-theoretic solutions of the Yang–Baxter equation

Abstract: Let $(X,r)$ be a set-theoretic solution of the YBE, that is $X$ is a set and $r\colon X\times X \to X\times X$ satisfies
$$(r \times \mathrm{id})\circ (\mathrm{id} \times r)\circ (r \times \mathrm{id}) = (\mathrm{id} \times r)\circ (r \times \mathrm{id})\circ (\mathrm{id} \times r)$$ on $X^{3}$. Write $r(x,y)=(\lambda_x (y), \rho_y (x))$, for $x,y\in X$. Gateva-Ivanova and Majid showed that the study of such solutions is determined by solutions $(M,r_M)$, where
$M=M(X,r) =\langle x\in X\mid xy=\lambda_x(y) \rho_y(x), \text{ for all } x,y\in X \rangle$
is the structure monoid of  $(X,r)$, and $r_M$ restricts to $r$ on $X^2$. For left non-degenerate solutions, i.e. all $\sigma_x$ are bijective, it has been shown that $M(X,r)$ is a regular submonoid of $A(X,r)\rtimes \mathcal{G}(X,r)$, where $\mathcal{G}(X,r)=\langle \lambda_x\mid x\in X\rangle$ is the permutation group of $(X,r)$, and
$A(X,r) =\langle x\in X \mid x\lambda_{x}(y) =\lambda_{x}(y) \lambda_{\sigma_{x}(y)}(\rho_{y}(x) \rangle$
is the derived monoid of $(X,r)$. It also is the structure monoid of the rack solution $(X,r’)$ with
$r'(x,y)=(y,\lambda_y\rho_{\lambda^{-1}_x(y)}(x)).$
This solution “encodes”  the relations determined by the map $r^{2} \colon X^{2} \to X^{2}$. The elements of $A=A(X,r)$ are normal, i.e.  $aA=Aa$ for all $a\in A$. It is this “richer structure” that has been exploited by several authors to obtain information on the structure monoid $M(X,r)$ and the structure algebra $kM(X,r)$.

In this talk  we report on some  recent investigations for arbitrary solutions, i.e. not necessarily left non-degenerate nor bijective.
This is joint work with F. Ced\’o and C. Verwimp.  We prove that there is a  unique $1$-cocycle $M(X,r)\to A(X,r)$ and we determine when this mapping is injective, surjective, respectively bijective. One then obtains a monoid homomorphism $M(X,r) \to A(X,r)\rtimes \langle \sigma_x \mid x\in X\rangle$. This mapping is injective when all $\sigma_x$ are injective. Further we determine the left cancellative congruence $\eta$ on $M(X,r)$ and show that $(X,r)$ induces a set-theoretic solution in $M(X,r)/\eta$ provided $(X,r)$ is left non-degenerate.

## Seminar (Operator Algebras)

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.

## 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.