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Colloquium - Ilijas Farah

Ilijas Farah
September 27, 2018
4:15PM - 5:15PM
Cockins Hall 240

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Add to Calendar 2018-09-27 16:15:00 2018-09-27 17:15:00 Colloquium - Ilijas Farah Title: Coarse spaces and uniform Roe algebras Speaker: Ilijas Farah (York University) Abstract: Coarse geometry is the study of large-scale properties of metric spaces. Roughly, two spaces are coarsely equivalent if their `large-scale structures' agree. The uniform Roe algebra $C*_u(X)$ is a norm-closed algebra of bounded linear operators on the Hilbert space $\ell_2(X)$. It is the algebra of all bounded linear operators on $\ell_2(X)$ that can be uniformly approximated by operators of `finite propagation'. The uniform Roe algebra is a coarse invariant of the space $X$. It includes $\ell_\infty(X)$ (as the algebra of all operators of zero propagation) and the algebra of compact operators. After introducing the basics of coarse spaces and uniform Roe algebras, we will consider the following questions: If the uniform Roe algebras of $X$ and $Y$ are isomorphic, when can we conclude that X and Y are coarsely equivalent? The uniform Roe corona is obtained by modding out the compact operators from $C^*_u(X)$. If the uniform Roe coronas of X and Y are isomorphic, when can we conclude that $C^*_u(X)$ and $C^*_u(Y)$ are isomorphic, or at least have large isomorphic corners? Under some additional assumptions on $X$ and $Y$ (the uniform local finiteness and a weakening of property A - A stands for `amenability'), [1] has a positive answer. The answer to question [2], even for uniformly locally finite spaces with property A, is quite surprising. These talks will be based on a joint work with B.M. Braga and A. Vignati. Colloquium URL: https://web.math.osu.edu/colloquium/ Cockins Hall 240 Department of Mathematics math@osu.edu America/New_York public

Title: Coarse spaces and uniform Roe algebras

Speaker: Ilijas Farah (York University)

Abstract: Coarse geometry is the study of large-scale properties of metric spaces. Roughly, two spaces are coarsely equivalent if their `large-scale structures' agree. The uniform Roe algebra $C*_u(X)$ is a norm-closed algebra of bounded linear operators on the Hilbert space $\ell_2(X)$. It is the algebra of all bounded linear operators on $\ell_2(X)$ that can be uniformly approximated by operators of `finite propagation'. The uniform Roe algebra is a coarse invariant of the space $X$. It includes $\ell_\infty(X)$ (as the algebra of all operators of zero propagation) and the algebra of compact operators. After introducing the basics of coarse spaces and uniform Roe algebras, we will consider the following questions:

  1. If the uniform Roe algebras of $X$ and $Y$ are isomorphic, when can we conclude that X and Y are coarsely equivalent?
  2. The uniform Roe corona is obtained by modding out the compact operators from $C^*_u(X)$. If the uniform Roe coronas of X and Y are isomorphic, when can we conclude that $C^*_u(X)$ and $C^*_u(Y)$ are isomorphic, or at least have large isomorphic corners?

Under some additional assumptions on $X$ and $Y$ (the uniform local finiteness and a weakening of property A - A stands for `amenability'), [1] has a positive answer. The answer to question [2], even for uniformly locally finite spaces with property A, is quite surprising.

These talks will be based on a joint work with B.M. Braga and A. Vignati.

Colloquium URL: https://web.math.osu.edu/colloquium/

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