Poisson boundary on full Fock space

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 strictly 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 $e$. We observe that the non-commutative Poisson boundary associated with $\Big (B\big (\mathcal {F}(\mathcal {H})\big ), P_{\omega }\Big )$ is $\sigma$-weak closure of the Cuntz algebra $\mathcal {O}_{\dim \mathcal {H}}$ generated by the right creation operators. We prove that the Poisson boundary is 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’ $S$ invariant and curiously they are type $III _{\lambda }$ factors with $\lambda$ belonging to a certain small class of algebraic numbers.