Categoria: 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.

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.

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

A correspondence between dynamical systems and separated graphs

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.

Laurent Cantier (Universitat Autònoma de Barcelona) delivered the talk:

Recovering K_* from the Cu_1 semigroup

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

Villadsen Algebras: Projections and Vector Bundles

Abstract:

Les àlgebres de Villadsen són un tipus de C*-àlgebres que van ser utilitzades per trobar contraexemples a la conjectura de Classificació d’Elliott. Per provar que la conjectura fallava van utilitzar que l’ordre de les projeccions sobre espais topologics s’associa a l’ordre entre els vector bundles d’aquests. Així, en les àlgebres de Villadsen s’utilitza fortament la teoria de vector bundles per tal de construir l’exemple dessitjat. En aquesta xerrada explicarem una mica la història de la classificació de C*-àlgebres, i donarem algunes pinzellades sobre com utilitzar la teoria de vector bundles i les classes de Chern al món de les C*-àlgebres.