Poisson boundary on full Fock space

TL;DR

This paper investigates the non-commutative Poisson boundary of a specific operator algebra on the full Fock space, revealing it to be an injective type III factor and classifying it via Connes' S-invariant, especially in finite dimensions.

Contribution

It provides a complete classification of the non-commutative Poisson boundary on full Fock space, identifying its type and invariants, which was previously unknown.

Findings

Poisson boundary is an injective factor of type III for any 

Finite-dimensional case yields classification via Connes S-invariant

Poisson boundary types are often _{\u03bb} factors with algebraic 

Abstract

This article is devoted to studying the non-commutative Poisson boundary associated with $\Big(B\big(\mathcal{F}(\mathcal{H})\big), P_{\omega}\Big)$ where $\mathcal{H}$ is a separable Hilbert space (finite or infinite-dimensional), $\dim \mathcal{H} > 1$, with an orthonormal basis $\mathcal{E}$, $B\big(\mathcal{F}(\mathcal{H})\big)$ 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 $B\big(\mathcal{F}(\mathcal{H})\big)$ defined by \begin{align*} P_{\omega}(x) = \sum_{e \in \mathcal{E}} \omega_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 $\Big(B\big(\mathcal{F}(\mathcal{H})\big), P_{\omega}\Big)$ 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 $III _{\lambda }$ factors with $\lambda$ belonging to a certain small class of algebraic numbers.