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SUMMARY:N. Christopher Phillips (University of Oregon and Fields Institute
  for Research in Mathematical Sciences)
DTSTART:20231025T190000Z
DTEND:20231025T200000Z
DTSTAMP:20260420T053456Z
UID:NYC-NCG/141
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NYC-NCG/141/
 ">The radius of comparison of C (X) is about half the covering dimension o
 f X</a>\nby N. Christopher Phillips (University of Oregon and Fields Insti
 tute for Research in Mathematical Sciences) as part of Noncommutative geom
 etry in NYC\n\n\nAbstract\nRecall that a C*-algebra $A$ has strict compari
 son of projections\nif whenever $p$ and $q$ are projections in matrix alge
 bras over $A$\,\nand $\\tau (p) < \\tau (q)$ for all tracial states $\\tau
 $ on $A$\,\nthen $p$ is Murray-von Neumann subequivalent to $q$.\nIn conne
 ction with the Elliott\nclassification program\, and because many simple C
 *-algebras have\nvery few projections\, this has been extended to comparis
 on of\ngeneral positive elements.\n(This will be explained in the talk.)\n
 Strict comparison holds\nfor unital stably finite classifiable simple C*-a
 lgebras.\nThe radius of comparison ${\\mathrm{rc}} (A)$ of a C*-algebra $A
 $\nis a numerical measure of the failure of strict comparison.\nIt is zero
  if strict comparison holds\,\nand in general is a not so well understood 
 kind of topological dimension.\n\nLet $X$ be a compact metric space.\nIt h
 as been known for some time that ${\\mathrm{rc}} (C (X))$\nis at most abou
 t half the covering dimension of $X$.\nIn 2013\, Elliott and Niu proved th
 at ${\\mathrm{rc}} (C (X))$ is\,\nup to an additive constant\,\nat least h
 alf the rational cohomological dimension of $X$.\nRecently\, we proved tha
 t\, up to a slightly worse additive constant\,\n${\\mathrm{rc}} (C (X))$ i
 s at least half the covering dimension of $X$\,\nwhich is sometimes much l
 arger.\nThis shows that ${\\mathrm{rc}} (A)$\, like stable rank\, roughly\
 ncorresponds to covering dimension\, not to rational or integral\ncohomolo
 gical dimension\, and not to some previously unknown dimension.\n
LOCATION:https://researchseminars.org/talk/NYC-NCG/141/
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