Communications in Mathematical Physics

, Volume 229, Issue 2, pp 209-227

First online:

Functors of White Noise Associated to Characters of the Infinite Symmetric Group

  • Marek Boz˙ejkoAffiliated withInstytut Matematyczny, Uniwersytet Wrocł awski, Plac Grunwałdzki 2/4, 50-384 Wrocław, Poland
  • , Mădălin Gut¸ăAffiliated withMathematisch Instituut, Katholieke Universiteit Nijmegen, Toernooiveld 1, 6526 ED Nijmegen, The Netherlands. E-mail: guta@sci.kun.nl

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Abstract

 The characters \(\) of the infinite symmetric group are extended to multiplicative positive definite functions \(\) on pair partitions by using an explicit representation due to Veršik and Kerov. The von Neumann algebra \(\) generated by the fields \(\) with f in an infinite dimensional real Hilbert space \(\) is infinite and the vacuum vector is not separating. For a family \(\) depending on an integer N< - 1 an ``exclusion principle'' is found allowing at most \(\) ``identical particles'' on the same state:
$$$$
The algebras \(\) are type \(\) factors. Functors of white noise \(\) are constructed and proved to be non-equivalent for different values of N.