Decomposable semigroups on C*-algebras and D-divisible dynamical maps
Krzysztof Szczygielski
TL;DR
The paper addresses the problem of characterizing generators of decomposable semigroups on general C*-algebras and, for Hilbert spaces, of D-divisible quantum dynamics on the trace-class predual $B_1(\mathcal{H})$. The main approach shows that a smoothly decomposable generator $L$ splits as $L=L_{1}+L_{2}$ with $L_{1}$ completely dissipative and $L_{2}$ decomposable, yielding GKSL-like forms in both unital and nonunital cases and extending to weak-* continuous von Neumann algebra settings; this generalizes prior matrix-based results (e.g. Szczygielski, 2023) to arbitrary Hilbert spaces. The results provide explicit generator structures, such as $L(A)= i[H,A] + \phi(A) - \frac{1}{2}\{\phi(\mathds{1}),A\}$ in the unital setting and $L(a)= \varphi(a) + (\tau\circ\psi)(a) + Ka + aK^{*}$ in the nonunital setting, along with a D-divisible form for time-dependent dynamics on $B_1(\mathcal{H})$: $\mathcal{L}_{t}(\rho) = -i[H_t,\rho] + \Phi_t(\rho) - \frac{1}{2}\{\Phi'_t(\mathds{1}),\rho\}$. These advances extend the Lindblad framework beyond complete positivity, offering a robust mathematical template for broader quantum evolutions and raising questions about the role of smooth decomposability and indecomposable generators.
Abstract
We analyze semigroups of decomposable maps on C*-algebras in context of the algebraic structure of associated infinitesimal generators. Case of von Neumann algebras, including $B(\mathcal{H})$ for $\mathcal{H}$ a Hilbert space, is also addressed. We then elaborate on D-divisible (decomposably divisible) dynamical maps on the Banach space of trace class operators. Our analysis extends earlier results on decomposable dynamical maps on matrix algebras (J. Phys. A: Math. Theor. 56 485202) and provides a partial generalization of the seminal work of Lindblad (Commun. Math. Phys. 48 119-130) on completely positive semigroups.