  The identification of the $*$--Lie algebra of the renormalized higher powers
of white noise (RHPWN) and the analytic continuation of the second quantized
Virasoro--Zamolodchikov--$w_{\infty} *$--Lie algebra of conformal field theory
and high-energy physics, was recently established in \cite{id} based on results
obtained in [1] and [2]. In the present paper we show how the RHPWN Fock
kernels must be truncated in order to be positive definite and we obtain a Fock
representation of the two algebras. We show that the truncated renormalized
higher powers of white noise (TRHPWN) Fock spaces of order $\geq 2$ host the
continuous binomial and beta processes.
