On von Neumann algebras generated by free Poisson random weights
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915227722842112 |
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| author | Yang, Zhiyuan |
| author_facet | Yang, Zhiyuan |
| contents | We study a generalization of free Poisson random measure by replacing the intensity measure with a n.s.f. weight $φ$ 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 $L^2(M,φ)$ and study the free Poisson von Neumann algebra $Γ(M,φ)$ 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 $φ(1)<\infty$, we show that $Γ(M,φ)$ can be decomposed into free product of other algebras. For a general weight $φ$, we prove that $ Γ(M,φ) $ is a factor if and only if $ φ(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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03087 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On von Neumann algebras generated by free Poisson random weights Yang, Zhiyuan Operator Algebras We study a generalization of free Poisson random measure by replacing the intensity measure with a n.s.f. weight $φ$ 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 $L^2(M,φ)$ and study the free Poisson von Neumann algebra $Γ(M,φ)$ 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 $φ(1)<\infty$, we show that $Γ(M,φ)$ can be decomposed into free product of other algebras. For a general weight $φ$, we prove that $ Γ(M,φ) $ is a factor if and only if $ φ(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. |
| title | On von Neumann algebras generated by free Poisson random weights |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2504.03087 |