# POISSON BOUNDARY ON FULL FOCK SPACE

## Article Type

Research Article

## Publication Title

Transactions of the American Mathematical Society

## Abstract

This article is devoted to studying the non-commutative Poisson boundary associated with (B(F(H)), Pω) where H is a separable Hilbert space (finite or infinite-dimensional), dim H > 1, with an orthonormal basis ℇ B(F(H)) is the algebra of bounded linear operators on the full Fock space F(H) defined over H, ω = {ωe : e ∈ E} is a sequence of strictly positive real numbers such that∑ e ωe = 1 and Pω is the Markov operator on B(F(H)) defined by [Formular Presented] where, for e ∈ E, le denotes the left creation operator associated with e. We observe that the non-commutative Poisson boundary associated with (B(F(H)), Pω)is σ-weak closure of the Cuntz algebra Odim H generated by the right creation operators. We prove that the Poisson boundary is an injective factor of type III for any choice of ω. Moreover, if H is finite-dimensional, we completely classify the Poisson boundary in terms of its Connes’ S invariant and curiously they are type IIIλ factors with λ belonging to a certain small class of algebraic numbers.

## First Page

5645

## Last Page

5668

## DOI

10.1090/tran/8684

## Publication Date

8-1-2022

## Recommended Citation

Bhat, B. V.Rajarama; Bikram, Panchugopal; De, Sandipan; and Rakshit, Narayan, "POISSON BOUNDARY ON FULL FOCK SPACE" (2022). *Journal Articles*. 3006.

https://digitalcommons.isical.ac.in/journal-articles/3006

## Comments

Open Access, Green