Covariant decomposable maps on C*-algebras and quantum dynamics

We characterize covariant positive decomposable maps between unital C*-algebras in terms of a dilation theorem, which generalizes a seminal result by H. Scutaru from Rep. Math. Phys. 16 (1):79-87, 1979. As a case study, we provide a certain characterization of the operator sum representation of maps on $\mathbb{M}_n (\mathbb{C})$, covariant with respect to the maximal commutative subgroup of $\mathrm{U}(n)$. A connection to quantum dynamics is established by specifying sufficient and necessary conditions for covariance of D-divisible (decomposably divisible) quantum evolution families, recently introduced in J. Phys. A: Math. Theor. 56 (2023) 485202.