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

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.