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

Manifolds of classical probability distributions and quantum density operators in infinite dimensions