On the product system of a completely positive semigroup

Given a W∗-continuous semigroup φ of unital, normal, completely positive maps of B(H), we introduce its continuous tensor product system Eφ. If α is a minimal dilation E0-semigroup of φ with Arveson product system F, then Eφ is canonically isomorphic to F. We apply this construction to a class of semigroups of B(L2(R)) arising from a modified Weyl–Moyal quantization of convolution semigroups of Borel probability measures on R2. This class includes the heat flow on the CCR algebra studied recently by Arveson. We prove that the minimal dilations of all such semigroups are completely spatial, and additionally, we prove that the minimal dilation of the heat flow is cocyle conjugate to the CAR/CCR flow of index two.