Classification of q-pure q-weight maps over finite dimensional Hilbert spaces

An $E_0$-semigroup of $B(H)$ is a one parameter strongly continuous semigroup of $\ast$-endomorphisms of $B(H)$ that preserve the identity. Every $E_0$-semigroup that possesses a strongly continuous intertwining semigroup of isometries is cocycle conjugate to an $E_0$-semigroup induced by the Bhat induction of a $CP$-flow over a separable Hilbert space $K$. We say an $E_0$-semigroup $\alpha$ is $q$-pure if the $CP$-subordinates $\beta$ of norm one (i.e. $\mathrm{\parallel }{\beta }_t(I)\mathrm{\parallel }=1$ and ${\alpha }_t-{\beta }_t$ is completely positive for all $t\ge 0$) are totally ordered in the sense that if $\beta$ and $\gamma$ are two $CP$-subordinates of $\alpha$ of norm one, then $\beta \geq \gamma$or$\gamma \geq \beta$. This paper shows how to construct and classify all $q$-pure $E_0$-semigroups induced by $CP$-flows over a finite-dimensional Hilbert space $K$ up to cocycle conjugacy.