## Poisson boundary on full Fock space

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- by B. V. Rajarama Bhat, Panchugopal Bikram, Sandipan De and Narayan Rakshit PDF
- Trans. Amer. Math. Soc.
**375**(2022), 5645-5668 Request permission

## 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.## References

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## Additional Information

**B. V. Rajarama Bhat**- Affiliation: Indian Statistical Institute, Stat-Math Unit, R V College Post, Bengaluru 560059, India
- MR Author ID: 314081
- ORCID: 0000-0002-4614-8890
- Email: bhat@isibang.ac.in
**Panchugopal Bikram**- Affiliation: School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar, An OCC of Homi Bhabha National Institute, Jatni-752050, India
- MR Author ID: 971245
- Email: bikram@niser.ac.in
**Sandipan De**- Affiliation: School of Mathematics and Computer Science, Indian Institute of Technology Goa, Farmagudi, Ponda-403401, Goa, India
- MR Author ID: 1122690
- ORCID: 0000-0003-0431-8633
- Email: sandipan@iitgoa.ac.in
**Narayan Rakshit**- Affiliation: Indian Statistical Institute, Stat-Math Unit, R V College Post, Bengaluru 560059, India
- MR Author ID: 1231896
- ORCID: 0000-0001-7756-8292
- Email: narayan753@gmail.com
- Received by editor(s): September 17, 2021
- Received by editor(s) in revised form: January 1, 2022
- Published electronically: May 23, 2022
- Additional Notes: The first author thanks J C Bose Fellowship of SERB (India) for financial support. The second author was supported by the grant CEFIPRA-6101-1 and the fourth author was supported by the NBHM (India) post-doctoral fellowship
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 5645-5668 - MSC (2020): Primary 46L10, 46L36; Secondary 46L40, 46L53
- DOI: https://doi.org/10.1090/tran/8684
- MathSciNet review: 4469232

Dedicated: Dedicated to Prof. V.S. Sunder