The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of $C^{*}$-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 $C^{*}$-algebra $\mathscr{A}$ which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states $\mathscr{S}$ of a possibly infinite-dimensional, unital $C^{*}$-algebra $\mathscr{A}$ is partitioned 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 the orbits through density operators on an infinite-dimensional, separable Hilbert space $\mathcal{H}$ are smooth, homogeneous Banach manifolds of $\mathscr{G}=\mathcal{GL}(\mathcal{H})$, and, when $\mathscr{A}$ admits a faithful tracial state $τ$ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through $τ$ is a smooth, homogeneous Banach manifold for $\mathscr{G}$.
Infinite-dimensional Frobenius Manifolds Underlying the Toda Lattice Hierarchy
根据Carlet et al.（2011）\cite{CDM}，我们构造了一类基于Toda格族的无穷维Frobenius流形，它们定义在原点和无穷远处可能具有高阶极点的亚纯函数对空间上。我们还显示了这些无限维Frobenius流形和有限维Frobenius流形的轨道空间上的扩展仿射Weyl群的类型$A$ Dubrovin和张之间的连接。
The mathematical role of (commutative and noncommutative) infinitesimal random walks over (commutative and noncommutative) riemannian manifolds in Quantum Physics
