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{\Large Department of Mathematics, BGU}

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{\Huge BGU Probability and Ergodic Theory  (PET) seminar}\\[0.2\baselineskip]

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\textbf{On} \emph{Thursday, April 28, 2022}
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\textbf{At} \emph{11:10 -- 12:00}
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\textbf{In} \emph{-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 Mean dimension of an action and the radius of comparison of its C*-algebra\par}
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\textsc{Abstract:}
For an action of a countable amenable group  $G$  on a compact metric
space  $X$, the mean dimension  $mdim (G, X)$ was introduced by
Lindenstrauss and Weiss, for reasons unrelated to $C^*$-algebras. The
radius of comparison  $rc (A)$  of a $C^*$-algebra  $A$  was introduced by
Toms, for use on $C^*$-algebras having nothing to do with dynamics.

A construction called the crossed product  $C^* (G, X)$  associates a
$C^*$-algebra to a dynamical system. 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 give the first general partial results
towards the direction  $rc ( C^* (G, X) ) \geq (1/2) mdim (G, X)$.
We don't get the exact conjectured bound, but we get nontrivial
results for many of the known examples of free minimal systems with
$mdim (G, X) > 0$.  The proof depends, among other things, on Cech
cohomology, and uses something we call the mean cohomological
independence dimension. Unlike the currently known results in the
other direction, it works for all choices of  $G$.

The talk will include something about the crossed product
construction; no previous knowledge of it will be assumed.

This is joint work with Ilan Hirshberg.








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