标题： 关于由自由泊松随机权生成的冯·诺依曼代数 显示英文标题

标题： On von Neumann algebras generated by free Poisson random weights

摘要： 我们研究了自由泊松随机测度的一种推广，通过用作用在冯·诺依曼代数$M$上的正常下半连续（n.s.f.）权重$\varphi$替换强度测度。我们利用$L^2(M,\varphi)$上的全Fo ck空间给出了自由泊松随机权重的一个显式构造，并研究了由这个随机权重生成的自由泊松冯·诺依曼代数$\Gamma(M,\varphi)$。这种构造可以看作是作用于左Hilbert代数上的自由泊松型函子，类似于Voiculescu为Hilbert空间定义的自由Gaussian函子。当$\varphi(1)<\infty$成立时，我们证明了$\Gamma(M,\varphi)$可以分解为其他代数的自由积。 对于一般的权值$\varphi$，我们证明了$ \Gamma(M,\varphi) $是因子当且仅当$ \varphi(1)\geq 1 $和$ M\neq \mathbb{C} $成立。 研究了次单位权值递减的完全正映射的二次量子化。 通过考虑左Hilbert代数的一个退化版本，我们也能将自由Araki-Woods代数作为退化左Hilbert代数自由Poisson代数的特例来处理。 我们表明，联合自由无限可分族（在一个迹概率空间中）的Lévy-Itô分解实际上可以解释为退化左Hilbert代数的一种分解。 最后，作为一种应用，我们在全Fock空间中给出了任意加法时间参数化自由Lévy过程的实现。 利用这一实现，我们证明了任意加法自由Lévy过程的过滤代数总是插值群因子，可能还带有一个额外的原子。 显示英文摘要

摘要： 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 $\Gamma(M,\varphi)$ 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 $\Gamma(M,\varphi)$ can be decomposed into free product of other algebras. For a general weight $\varphi$, we prove that $ \Gamma(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 L\'{e}vy-It\^o 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 L\'{e}vy process as unbounded operators in a full Fock space. Using this realization, we show that the filtration algebras of any additive free L\'{e}vy process are always interpolated group factors with a possible additional atom.