Categoria: Seminars

Publicat a

Seminar (Operator Algebras)

Eduard Vilalta (Universitat Autònoma de Barcelona)

The range problem for the Cuntz semigroup of AI-algebras

Abstract:

A C*-algebra A is said to be a (separable) AI-algebra if A is isomorphic to an inductive limit of the form $\lim\limits_n (C[0,1]\otimes M_n)$ with $F_n$ a finite dimensional C*-algebra for every n. Whenever A is unital and commutative, A is isomorphic to C(X) with X an inverse limit of finite disjoint copies of unit intervals.

In this 2-session talk, we will study the range problem for the Cuntz semigroup of AI-algebras. That is, we will study whether or not one can determine a natural set of properties that an abstract Cuntz semigroup must satisfy in order to be isomorphic to the Cuntz semigroup of an AI-algebra.

During the first part of the talk, we will focus on unital commutative AI-algebras. In this case, one is able to solve the range problem for this class, thus giving a list of properties that an abstract Cuntz semigroup S satisfies if and only if S is isomorphic to the Cuntz semigroup of such an algebra. In order to prove this result, we first introduce the notion of almost chainable spaces and prove that a compact metric space X is almost chainable if and only if C(X) is an AI-algebra. We also characterize when S is isomorphic to the Cuntz semigroup of lower-semicontinuous functions $X\to 0,1,\dots,\infty$ for some T1-space X. The results in this first session will appear in [4].

In the second session, we will present a local characterization for the Cuntz semigroup of any AI-algebra resembling Shen’s local characterization of dimension groups[3], later used in the celebrated Effros-Handelman-Shen theorem[2]. One of the key features in the proof of our result will be the notion of Cauchy sequences for Cu-morphisms (with respect to the distance introduced in [1]) and the fact that, under the right assumptions, they have a unique limit; see [5].

[1] Ciuperca, A. and Elliott, G. “A remark on invariants for C*-algebras of stable rank one”, Int. Math. Res. Not. IMRN(2008)
[2] Effros, E. G. and Handelman, D. E. and Shen, C. L. “Dimension groups and their affine representations”, Amer. J. Math.102(1980), 385–407.
[3] Shen, C. L. “On the classification of the ordered groups associated with the approximately finite dimensional C*-algebras” ,Duke Math. J.46(1979), 613–633.
[4] Vilalta, E. “The Cuntz semigroup of unital commutative AI-algebras”, in preparation.
[5] Vilalta, E. “A local characterization for the Cuntz semigroup of AI-algebras”, (preprint) arXiv:2102.13557 [math.OA]

Publicat a

Seminar (Ring Theory)

Giovanna Le Gros (Università di Padova)

Enveloping and n-tilting classes over commutative rings

Abstract: In this talk, we will discuss approximations by tilting classes over commutative rings, where tilting classes are generated by infinitely generated tilting modules. Recently, all the commutative rings over which 1-tilting classes are enveloping were classified. This used the classification of 1-tilting classes over commutative rings by faithful finitely generated Gabriel topologies proved by Hrbek. This classification was extended to the general tilting case by Hrbek-Stovicek, where instead the correspondence is with suitable finite sequences of Gabriel topologies. In this talk we will discuss some results toward the classification of tilting cotorsion pairs that provide approximations, using Hrbek and Stovicek’s classification. In particular, we will discuss how considering an induced tilting class in suitable factor rings retains useful properties of the original tilting class and their approximations.
This talk is based on current work with Dolors Herbera.

Publicat a

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.

Publicat a

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.

Publicat a

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.

Publicat a

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.