Generalization of the decomposable generator characterization to infinite-dimensional operator algebras B(H)

Determine whether the structural characterization Lt = -[Ht, ·] + Φt - Φt(I) for generators of D-divisible (decomposably divisible) quantum evolution families on Mn(C), with Ht Hermitian and Φt a decomposable linear map Mn(C) → Mn(C), admits a direct generalization to the infinite-dimensional setting B(H); specifically, establish whether generators of decomposable, norm-continuous dynamical semigroups on B(H) can be written in the form -[H, ·] + Ψ - Ψ(I) for some Hermitian H ∈ B(H) and a decomposable linear map Ψ : B(H) → B(H).

Background

The paper introduces D-divisible (decomposably divisible) quantum evolution families on Mn(C) and states that, under regularity, their generators admit a structure Lt = -[Ht, ·] + Φt - Φt(I) with Ht Hermitian and Φt decomposable. This provides a finite-dimensional analogue of Lindblad-type generator forms beyond complete positivity.

The authors note that this characterization currently applies only to finite-dimensional C*-algebras and explicitly state that it is unknown whether an analogous structure holds for arbitrary infinite-dimensional operator algebras B(H). Establishing such a result would significantly extend Lindblad’s framework to decomposable dynamics in infinite dimensions.