E-dilation of strongly commuting CP-semigroups (the nonunital case)

In a previous paper, we showed that every strongly commuting pair of CP_0-semigroups on a von Neumann algebra (acting on a separable Hilbert space) has an E_0-dilation. In this paper we show that if one restricts attention to the von Neumann algebra B(H) then the unitality assumption can be dropped, that is, we prove that every pair of strongly commuting CP-semigroups on B(H) has an E-dilation. The proof is significantly different from the proof for the unital case, and is based on a construction of Ptak from the 1980's designed originally for constructing a unitary dilation to a two-parameter contraction semigroup.