"POISSON BOUNDARY ON FULL FOCK SPACE" by B. V.Rajarama Bhat, Panchugopal Bikram et al.
 

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

Comments

Open Access, Green

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