Dilations, Product Systems, and Weak Dilations

We generalize Bhat's construction of product systems of Hilbert spaces from $E_0$-semigroups on $B(H)$ for some Hilbert space $H$ to the construction of product systems of Hilbert modules from $E_0$-semigroups on $B^a(E)$ for some Hilbert module $E$. As a byproduct we find the representation theory for $B^a(E)$ if $E$ has a unit vector. We establish a necessary and sufficient criterion for the conditional expectation generated by the unit vector to define a weak dilation of a $CP$-semigroup in the sense of [1]. Finally, we also show that white noises on general von Neumann algebras in the sense of [2] can be extended to white noises on a Hilbert module.