Quantized convolution semigroups

Abstract

Abstract. We describe our construction of a continuous tensor product system in the sense of Arveson for a general W * -continuous completely positive semigroup of B(H) (H separable). This product system is canonically isomorphic to the product system of the minimal dilation E 0 -semigroup. We use this construction to show that contrary to previous speculation the minimal dilations of all quantized convolution semigroups are completely spatial. This class includes the heat flow on the CCR algebra studied recently by Arveson, which we show is cocycle conjugate to a CAR/CCR flow of index two. The analysis of these examples also involves Lévy processes and their stochastic area processes.Additionally, we prove the following fact: given a product system F and a type I product system E, if there is a bijection ψ : E → F , not necessarily measurable, that preserves fibers and multiplication and ψ is unitary fiberwise, then E and F are isomorphic.