If you take the Fock space $\mathcal F_P$ associated to a polarization of $H$, as in [W, page 480, line 7], and consider the CAR algebra (=algebra generated by creation and annihilation operators) generated by a subspace $K\subset H$, as in [W, page 497, line 29], then the von Neumann algebra this generates is (hyperfinite) of type $III_1$ [W, Corollary on the bottom of page 500].
Reference:
[W] Wassermann, Operator algebras and conformal field theory
