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:<br> [W] Wassermann, <i>[Operator algebras and conformal field theory][1]</i>

  [1]: http://pages.uoregon.edu/njp/wass.pdf