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Question on Dixmier trace

  1. Mar 25, 2010 #1
    if the dixmier trace of an operator A is defined as

    [tex] \frac{\sum_{i=0}^{N}\mu_{i}}{logN} [/tex] in the limit N-->oo (infinity)

    where the eigenvalues are ordered in decreasing order

    would not it better to be defined as

    [tex] \frac{\sum_{i=0}^{N}\mu_{i}^{s}}{logN} [/tex]

    here 's' is a parameter to be FIXED mathematically so

    [tex] \sum_{i=0}^{N}\mu_{i}^{s}=T.log[/tex] then 'T' is just the Dixmier trace

    the idea is to define the number 's' to be real of complex so the zeta function of operator A

    [tex] Z=Tr(A^{s}) [/tex] has a pole or a logarithmic divergence , then the normal definition of dixmier trace is recovered when s=1
     
  2. jcsd
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