The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of $$C^{*}$$ C ∗ -algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given $$C^{*}$$ C ∗ -algebra $$\mathscr {A}$$ A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states $$\mathscr {S}$$ S of a possibly infinite-dimensional, unital $$C^{*}$$ C ∗ -algebra $$\mathscr {A}$$ A is partitioned into the disjoint union of the orbits of an action of the group $$\mathscr {G}$$ G of invertible elements of $$\mathscr {A}$$ A . Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space $$\mathcal {H}$$ H are smooth, homogeneous Banach manifolds of $$\mathscr {G}=\mathcal {GL}(\mathcal {H})$$ G = GL ( H ) , and, when $$\mathscr {A}$$ A admits a faithful tracial state $$\tau $$ τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through $$\tau $$ τ is a smooth, homogeneous Banach manifold for $$\mathscr {G}$$ G .