On von Neumann algebras generated by free Poisson random weights

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 $\Gamma(M,\varphi)$ 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.