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SUMMARY:N. Christopher Phillips (University of Oregon)
DTSTART:20240514T190000Z
DTEND:20240514T200000Z
DTSTAMP:20260420T053237Z
UID:NYC-NCG/144
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NYC-NCG/144/
 ">Minicourse: An invitation to mean dimension of a dynamical system   and 
 the radius of comparison of its crossed product\, II</a>\nby N. Christophe
 r Phillips (University of Oregon) as part of Noncommutative geometry in NY
 C\n\n\nAbstract\nThe purpose of this minicourse is to explain the backgrou
 nd\n(including the terms below) and some progress towards the following co
 njecture\, relating topological dynamics to the structure of the crossed p
 roduct $C^*$-algebra.\n\nLet $G$ be a countable amenable group\, let $X$ b
 e a compact metrizable space\,\nand let $T$ be an action of $G$ on $X$. Th
 e mean dimension $mdim ~(T)$ is a \npurely dynamical invariant\, designed 
 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 $A$ is 
 a numerical measure of failure of comparison\nin the Cuntz semigroup of $A
 $\, a generalization of unstable K-theory.\nIt was introduced to distingui
 sh $C^*$-algebras having no connection\nwith dynamics. The conjecture asse
 rts that if $T$ is free and minimal\,\nthen $rc ~(C^* (G\, X\, T)) = \\fra
 c{1}{2} ~mdim ~(T)$. The inequality\n$rc ~(C^* (G\, X\, T)) \\leq \\frac{1
 }{2} ~mdim ~(T)$ is known for \n$G = {\\mathbb{Z}}^n$\, and progress towar
 ds the inequality\n$rc ~(C^* (G\, X\, T)) \\geq \\frac{1}{2} ~mdim ~(T)$ h
 as been made for the known \nclasses of examples of free minimal actions w
 ith nonzero mean dimension\,\nfor any countable amenable group $G$. The em
 phasis 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
 Lecture 2.\n\nThis lecture will be mainly about comparison in $C^*$-algebr
 as.\nWe will describe comparison properties\, first for projections and th
 en for positive elements.\nThen we define the radius of comparison\, and s
 how how it is related to ``noncommutative dimension''.\n
LOCATION:https://researchseminars.org/talk/NYC-NCG/144/
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