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SUMMARY:N. Christopher Phillips (University of Oregon)
DTSTART:20240521T190000Z
DTEND:20240521T200000Z
DTSTAMP:20260420T053057Z
UID:NYC-NCG/145
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NYC-NCG/145/
 ">Minicourse:  An invitation to mean dimension of a dynamical system   and
  the radius of comparison of its crossed product\, III</a>\nby N. Christop
 her Phillips (University of Oregon) as part of Noncommutative geometry in 
 NYC\n\n\nAbstract\nThe purpose of this minicourse is to explain the backgr
 ound\n(including the terms below) and some progress towards the following 
 conjecture\, relating topological dynamics to the structure of the crossed
  product $C^*$-algebra.\n\nLet $G$ be a countable amenable group\, let $X$
  be a compact metrizable space\,\nand let $T$ be an action of $G$ on $X$. 
 The mean dimension $mdim ~(T)$ is a \npurely dynamical invariant\, designe
 d so that the mean dimension of the shift \non $([0\, 1]^d)^G$ is equal to
  $d$. The radius of comparison $rc ~(A)$ of a \nunital $C^*$-algebra $A$ i
 s a numerical measure of failure of comparison\nin the Cuntz semigroup of 
 $A$\, a generalization of unstable K-theory.\nIt was introduced to disting
 uish $C^*$-algebras having no connection\nwith dynamics. The conjecture as
 serts that if $T$ is free and minimal\,\nthen $rc ~(C^* (G\, X\, T)) = \\f
 rac{1}{2} ~mdim ~(T)$. The inequality\n$rc ~(C^* (G\, X\, T)) \\leq \\frac
 {1}{2} ~mdim ~(T)$ is known for \n$G = {\\mathbb{Z}}^n$\, and progress tow
 ards the inequality\n$rc ~(C^* (G\, X\, T)) \\geq \\frac{1}{2} ~mdim ~(T)$
  has been made for the known \nclasses of examples of free minimal actions
  with nonzero mean dimension\,\nfor any countable amenable group $G$. The 
 emphasis will be on the inequality\n$rc ~(C^* (G\, X\, T)) \\geq \\frac{1}
 {2} ~mdim ~(T)$\;\nthe results there are joint work with Ilan Hirshberg.\n
 \n\nLecture 3.\n\nIn this lecture\, we state some known results towards th
 e conjecture \n$rc ~(C^* (G\, X\, T)) = \\frac{1}{2} ~mdim ~(T)$\,\nand sa
 y something about the ideas which go into the results\ntowards the inequal
 ity $rc ~(C^* (G\, X\, T)) \\geq \\frac{1}{2} ~mdim ~(T)$.\n
LOCATION:https://researchseminars.org/talk/NYC-NCG/145/
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