Decomposable semigroups on C*-algebras and D-divisible dynamical maps

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.

💡 Research Summary

The paper investigates semigroups of decomposable maps on C*-algebras and extends the classical Lindblad theory of completely positive (CP) dynamical semigroups to a broader class of maps that are merely decomposable (i.e., sums of a CP map and a co‑CP map). After recalling Lindblad’s seminal result—that a uniformly continuous one‑parameter semigroup (e^{tL})_{t≥0} of unital CP maps on a unital C*-algebra A is generated by a completely dissipative ‑map L satisfying L(1)=0—the author introduces the notion of a decomposable map D(A). A map φ belongs to D(A) if it can be written φ=φ₁+φ₂ with φ₁∈CP(A) and φ₂∈coCP(A). Using Stinespring‑Størmer dilation theorems, any φ∈D(A) admits a representation φ(a)=Vπ(a)V where π is a Jordan morphism (the sum of a *‑homomorphism and a *‑anti‑homomorphism) and V is a bounded operator into a larger Hilbert space.

The core of the work consists of three main theorems for unital semigroups and their weak‑* continuous analogues on von Neumann algebras. Theorem 1 shows that if a uniformly continuous semigroup (e^{tL}) is “smoothly decomposable” – i.e., for each t≥0 there exist smooth families ϕ_t∈CP(A) and ψ_t∈CP(A) such that e^{tL}=ϕ_t+τ∘ψ_t (τ denotes transposition) – then the generator L splits as L=L₁+L₂ with L₁∈Dis_∞(A) (completely dissipative) and L₂∈coCP(A). Moreover L₁(1)≤0 and L₂(1)≤0, guaranteeing L(1)=0. Conversely, given any L₁∈Dis_∞(A) and L₂∈coCP(A), the Trotter product formula yields a uniformly continuous semigroup e^{t(L₁+L₂)} that is decomposable; if additionally L₁(1)+L₂(1)=0 the semigroup is unital. Theorem 2 lifts these statements to the setting of a von Neumann algebra A, requiring the generators to be weak‑* continuous (denoted by the superscript σ). Theorem 3 refines the structure further: when L₁∈Dis_∞^σ(A), L₂∈coCP^σ(A) and L₁(1)+L₂(1)=0, there exist a self‑adjoint element H∈A and a decomposable map φ∈D^σ(A) such that

 L(a)=i