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

*The real rank of uniform Roe algebras I*

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*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.