Computation of some examples of Brown's spectral measure in free probability

Volume 90 / 2001

Philippe Biane, Franz Lehner Colloquium Mathematicum 90 (2001), 181-211 MSC: Primary 22D25, 46L54; Secondary 15A52, 43A05, 60J15. DOI: 10.4064/cm90-2-3


We use free probability techniques to compute spectra and Brown measures of some non-hermitian operators in finite von Neumann algebras. Examples include $u_n+u_{\infty }$ where $u_n$ and $u_{\infty }$ are the generators of ${\mathbb Z}_n$ and ${\mathbb Z}$ respectively, in the free product ${\mathbb Z}_n\mathbin {*}{\mathbb Z}$, or elliptic elements of the form $S_{\alpha }+iS_{\beta }$ where $S_\alpha $ and $S_\beta $ are free semicircular elements of variance $\alpha $ and $\beta $.


  • Philippe BianeCNRS
    45 rue d'Ulm
    75005 Paris, France
  • Franz LehnerInstitut für Analysis und Numerik
    Johannes Kepler Universität Linz
    4040 Linz, Austria
    Équipe d'Analyse
    Université Paris 6
    Boîte 186, Tour 46-0
    4, place Jussieu
    75252 Paris Cedex 05, France

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