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{\Large Department of Mathematics, BGU}

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{\Huge Colloquium}\\[0.2\baselineskip]

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\textbf{On} \emph{Tuesday, April 12, 2022}
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\textbf{At} \emph{14:30 -- 15:30}
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\textbf{In} \emph{Math -101}

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{\large\scshape Chris Phillips 
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  (University of Oregon)
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will talk about
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{\Large\bfseries Relations between dynamics and C*-algebras: Mean dimension and radius of comparison\par}
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\textsc{Abstract:}
This is joint work with Ilan Hirshberg.

For an action of an amenable group  G  on a compact metric space  X, the mean dimension  mdim (G, X) was introduced by Lindenstrauss and Weiss. It is designed so that the mean dimension of the shift on ({[}0, 1{]}\^{}d)\^{}G  is  d.  Its motivation was unrelated to C*-algebras.

The radius of comparison  rc (A)  of a C*-algebra  A  was introduced by Toms to distinguish counterexamples in the Elliott classification program. The algebras he used have nothing to do with dynamics.

A construction called the crossed product  C\^{}* (G, X)  associates a C\emph{-algebra to a dynamical system. Despite the apparent lack of connection between these concepts, there is significant evidence for the conjecture that  rc ( C\^{}} (G, X) ) = (1/2) mdim (G, X)  when the action is free and minimal. We will explain the concepts above; no previous knowledge of mean dimension, C\emph{-algebras, or radius of comparison will be assumed. Then we describe some of the evidence. In particular, we give the first general partial results towards the direction  rc ( C\^{}} (G, X) ) \textbackslash{}geq (1/2) mdim (G, X).  We don't get the exact conjectured bound, but we get nontrivial results for many of the known examples of free minimal systems with  mdim (G, X) \textgreater{} 0.








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