Completely positive semigroups and their product systems

This PhD. thesis is concerned with the minimal dilation of semigroups of completely positive maps on B(H) ( H a separable Hilbert space) to semigroups of endomorphisms.; The first part of this thesis introduces the construction of the continuous tensor product system of a completely positive semigroup. This product system has as fibers the inductive limits of appropriately selected tensor products of the metric operators spaces (in the sense of W. Arveson) associated with the completely positive semigroup. This construction relates naturally to previous work of W. Arveson that associates a product system to an E0-semigroup. Indeed, we prove that if alpha is an E0-semigroup with product system E, and alpha is a minimal dilation of a completely positive semigroup &phis;, with product system F arising from our construction, then we have a canonical isomorphism E ≃ F. In particular, all cocycle conjugacy invariants, and the type of the minimal dilation of &phis; can be studied directly through F.; The second part of the thesis considers the class of completely positive semigroups of B(L2( R2 )) which we call quantized convolution semigroups. These semigroups arise from a modified Weyl-Moyal quantization of convolution semigroups of Borel probability measures on R2 . The heat flow introduced by W. Arveson is an example of a quantized convolution semigroup, corresponding to the Gaussian convolution semigroup on the plane. We prove that the product system of a quantized convolution semigroup is isomorphic to the product system associated to a related R2 -valued Levy process. As a corollary, we show that all quantized convolution semigroups are spatial, therefore settling in the negative a pre-existing conjecture to the contrary. Additionally, we provide the first instance of the concrete identification and classification of a completely positive semigroup with unbounded generator, by showing that the heat flow is cocycle conjugate to a CAR/CCR flow of index two.