Structure of block quantum dynamical semigroups and their product systems

Abstract

W. Paschke's version of Stinespring's theorem associates a Hilbert $C^{\ast }$-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a $C^{\ast }$-algebra $\mathcal{A}$ one may associate an inclusion system $E=(E_t)$ of Hilbert $\mathcal{A}$-$\mathcal{A}$-modules with a generating unit $\xi =({\xi }_t)$. Suppose $\mathcal{B}$ is a von Neumann algebra, consider $M_2(\mathcal{B})$, the von Neumann algebra of $2\times 2$ matrices with entries from $\mathcal{B}$. Suppose $({\Phi }_t{)}_{t\ge 0}$ with ${\Phi }_t=\left(\begin{array}{ccc}{\varphi }_t^1& {\psi }_t{\psi }_t^{\ast }& {\varphi }_t^2\end{array}\right),$ is a QDS on $M_2(B)$ which acts block-wise and let $(E_t^i{)}_{t\ge 0}$ be the inclusion system associated to the diagonal QDS $({\varphi }_t^i{)}_{t\ge 0}$ with the generating unit $({\xi }_t^i{)}_{t\ge 0},i=1,2.$ It is shown that there is a contractive (bilinear) morphism $T=(T_t{)}_{t\ge 0}$ from $(E_t^2{)}_{t\ge 0}$ to $(E_t^1{)}_{t\ge 0}$ such that ${\psi }_t(a)=⟨{\xi }_t^1,T_{t}a{\xi }_t^2⟩$ for all $a\in \mathcal{B}.$ We also prove that any contractive morphism between inclusion systems of von Neumann $\mathcal{B}$-$\mathcal{B}$-modules can be lifted as a morphism between the product systems generated by them. We observe that the $E_0$-dilation of a block quantum Markov semigroup (QMS) on a unital $C^{\ast }$-algebra is again a semigroup of block maps.

17 pages. Significant Changes in Section 3

References in corpus (1)

• Inclusion systems and amalgamated products of product systems