On von Neumann algebras generated by free Poisson random weights
Zhiyuan Yang
TL;DR
This paper develops a robust free Poisson framework for von Neumann algebras with n.s.f. weights by constructing the centered free Poisson random weight X(h) on the full Fock space over L^2(M,φ) and the associated free Poisson algebra Γ(M,φ). It establishes a full operator-algebraic picture: a functorial free Poisson construction, a second-quantization mechanism for weight-decreasing completely positive maps, a decomposition theory that reduces finite-weight cases to free products with L(Z), and a detailed factoriality/type classification of Γ(M,φ). The work further connects free Poisson algebras to degenerate left Hilbert algebras, enabling a unified treatment of free Araki–Woods algebras and semicircular/Cauchy-type components, and provides Fock-space realizations of freely infinitely divisible distributions via Lévy-Itô decompositions. As applications, the authors derive filtration results for free Lévy processes, showing interpolation properties for filtration algebras and establishing a framework to realize additive free Lévy processes as unbounded operators in a free Poisson Fock space. Overall, the paper broadens the scope of free probability in operator algebras by blending left Hilbert algebra techniques with Poisson-type second quantization and Lévy-process realizations, yielding sharp structural results and new avenues for analyzing noncommutative stochastic processes.
Abstract
We study a generalization of free Poisson random measure by replacing the intensity measure with a n.s.f. weight $\varphi$ on a von Neumann algebra $M$. We give an explicit construction of the free Poisson random weight using full Fock space over the Hilbert space $L^2(M,\varphi)$ and study the free Poisson von Neumann algebra $Γ(M,\varphi)$ generated by this random weight. This construction can be viewed as a free Poisson type functor for left Hilbert algebras similar to Voiculescu's free Gaussian functor for Hilbert spaces. When $\varphi(1)<\infty$, we show that $Γ(M,\varphi)$ can be decomposed into free product of other algebras. For a general weight $\varphi$, we prove that $ Γ(M,\varphi) $ is a factor if and only if $ \varphi(1)\geq 1 $ and $ M\neq \mathbb{C} $. The second quantization of subunital weight decreasing completely positive maps are studied. By considering a degenerate version of left Hilbert algebras, we are also able to treat free Araki-Woods algebras as special cases of free Poisson algebras for degenerate left Hilbert algebras. We show that the Lévy-Itô decomposition of a jointly freely infinitely divisible family (in a tracial probability space) can in fact be interpreted as a decomposition of a degenerate left Hilbert algebra. Finally, as an application, we give a realization of any additive time-parameterized free Lévy process as unbounded operators in a full Fock space. Using this realization, we show that the filtration algebras of any additive free Lévy process are always interpolated group factors with a possible additional atom.