Peripheral Poisson Boundary on Full Fock space

TL;DR

This paper investigates the structure of the peripheral Poisson boundary generated by eigenvectors of a specific completely positive map on the algebra of bounded operators over a full Fock space, revealing its properties and relation to the Poisson boundary.

Contribution

It characterizes the peripheral Poisson boundary for a class of maps induced by creation operators on full Fock space, linking it to the Poisson boundary in this setting.

Findings

The peripheral Poisson boundary forms a C*-algebra generated by eigenvectors.

The boundary's structure is explicitly described for maps induced by creation operators.

The relationship between the peripheral and Poisson boundaries is clarified.

Abstract

The operator space generated by peripheral eigenvectors of a unital normal completely positive map $P$ on a von Neumann algebra has a C*-algebra structure. This C*-algebra is known as the \textit{peripheral Poisson boundary} of $P$. For a separable Hilbert space $H$, consider the full fock space defined over $H$. In this paper, we study the peripheral Poisson boundary of the completely positive map, induced by left creation operators of the basis vectors of $H$, on $B(\mcal F(H))$ and explore its behavior with respect to the Poisson boundary.