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2 results

  • Remarks on free mutual information and orbital free entropy

  • Masaki Izumi, Yoshimichi Ueda
  • Journal:
    Nagoya Mathematical Journal / Volume 220 / December 2015
    Published online by Cambridge University Press:
    11 January 2016, pp. 45-66
    Print publication:
    December 2015
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  • The present notes provide a proof of i* (ℂP + ℂ(I - P); ℂQ + ℂ(I - Q)) = – χorb(P,Q) for any pair of projections P,Q with τ(P) = τ(Q) = 1/2. The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.

  • Some irreducible free group representations in which a linear combination of the generators has an eigenvalue

  • Part of
  • William L. Paschke
  • Journal:
    Journal of the Australian Mathematical Society / Volume 72 / Issue 2 / April 2002
    Published online by Cambridge University Press:
    09 April 2009, pp. 257-286
    Print publication:
    April 2002
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  • We construct irreducible unitary representations of a finitely generated free group which are weakly contained in the left regular representation and in which a given linear combination of the generators has an eigenvalue. When the eigenvalue is specified, we conjecture that there is only one such representation. The representation we have found is described explicitly (modulo inversion of a certain rational map on Euclidean space) in terms of a positive definite function, and also by means of a quasi-invariant probability measure on the combinatorial boundary of the group.