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Poisson Boundary of the Dual of SU q (n)

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Abstract

We prove that for any non-trivial product-type action α of SU q (n) (0<q<1) on an ITPFI factor N, the relative commutant is isomorphic to the algebra L ∞() of bounded measurable functions on the quantum flag manifold . This is equivalent to the computation of the Poisson boundary of the dual discrete quantum group . The proof relies on a connection between the Poisson integral and the Berezin transform. Our main technical result says that a sequence of Berezin transforms defined by a random walk on the dominant weights of SU(n) converges to the identity on the quantum flag manifold.

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto, 606-8502, Japan

    Masaki Izumi

  2. Mathematics Institute, University of Oslo, PB 1053, Blindern, Oslo, 0316, Norway

    Sergey Neshveyev & Lars Tuset

Authors
  1. Masaki Izumi
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  2. Sergey Neshveyev
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  3. Lars Tuset
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Correspondence to Masaki Izumi.

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Communicated by Y. Kawahigashi

Supported by JSPS.

Partially supported by the Norwegian Research Council.

Supported by the SUP-program of the Norwegian Research Council.

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Izumi, M., Neshveyev, S. & Tuset, L. Poisson Boundary of the Dual of SU q (n). Commun. Math. Phys. 262, 505–531 (2006). https://doi.org/10.1007/s00220-005-1439-x

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