Narrow Results

Cipriani and Sauvageot have shown that for any $L^2$-generator $L^{(2)}$ of a tracially symmetric quantum Markov semigroup on a C*-algebra ${𝒜}$ there exists a densely defined derivation $\delta$ from ${𝒜}$ to a Hilbert bimodule $H$ such that $L^{(2)}={\delta }^{\ast }\circ \overline{\delta }$. Here, we show that this construction of a derivation can in general not be generalized to quantum Markov semigroups that are symmetric with respect to a non-tracial state. In particular, we show that all derivations to Hilbert bimodules can be assumed to have a concrete form, and then we use this form to show that in the finite-dimensional case the existence of such a derivation is equivalent to the existence of a positive matrix solution of a system of linear equations. We solve this system of linear equations for concrete examples using Mathematica to complete the proof.