Poisson boundary on full Fock space

Abstract

This article is devoted to studying the non-commutative Poisson boundary associated with $\left(B\right(\mathcal{F}(\mathcal{H})),P_{\omega })$ where $\mathcal{H}$ is a separable Hilbert space (finite or infinite-dimensional), $\dim \mathcal{H}>1$, with an orthonormal basis $\mathcal{E}$, $B(\mathcal{F}(\mathcal{H}))$ is the algebra of bounded linear operators on the full Fock space $\mathcal{F}(\mathcal{H})$ defined over $\mathcal{H}$, $\omega =\{{\omega }_e:e\in \mathcal{E}\}$ is a sequence of positive real numbers such that ${\sum }_e{\omega }_e=1$ and $P_{\omega }$ is the Markov operator on $B(\mathcal{F}(\mathcal{H}))$ defined by \begin{align*} P_ω(x) = \sum_{e \in \mathcal{E}} ω_e l_e^* x l_e, \ x \in B\big(\mathcal{F}(\mathcal{H})\big), \end{align*} where, for $e\in \mathcal{E}$, $l_e$ denotes the left creation operator associated with $e$. The non-commutative Poisson boundary associated with $\left(B\right(\mathcal{F}(\mathcal{H})),P_{\omega })$ turns out to be an injective factor of type $III$ for any choice of $\omega$. Moreover, if $\mathcal{H}$ is finite-dimensional, we completely classify the Poisson boundary in terms of its Connes $S$-invarinat and curiously they are type $II{I}_{\lambda }$ factors with $\lambda$ belonging to a certain small class of algebraic numbers.

Substantial revision has been made, proofs of some results are rewritten, one section is removed. To appear in Trans. Amer. Math. Soc