Doob equivalence and non-commutative peaking for Markov chains

Xinxin Chen; Adam Dor-On; Langwen Hui; Christopher Linden; Yifan Zhang

doi:10.4171/JNCG/444

Doob equivalence and non-commutative peaking for Markov chains

Xinxin Chen, Adam Dor-On, Langwen Hui, Christopher Linden, Yifan Zhang

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize the coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the noncommutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matrix. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras.

Original language	English
Pages (from-to)	1469-1484
Number of pages	16
Journal	Journal of Noncommutative Geometry
Volume	15
Issue number	4
DOIs	https://doi.org/10.4171/JNCG/444
State	Published - 2021
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2021 European Mathematical Society Published by EMS Press This work is licensed under a CC BY 4.0 license

Keywords

Doob equivalence
Harmonic functions
Liouville property
Non-commutative peaking
Rigidity
Stochastic matrices
Tensor algebras

ASJC Scopus subject areas

Algebra and Number Theory
Mathematical Physics
Geometry and Topology

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10.4171/JNCG/444

Cite this

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abstract = "In this paper, we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize the coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the noncommutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matrix. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras.",

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AU - Dor-On, Adam

AU - Hui, Langwen

AU - Linden, Christopher

AU - Zhang, Yifan

PY - 2021

Y1 - 2021

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AB - In this paper, we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize the coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the noncommutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matrix. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras.

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KW - Harmonic functions

KW - Liouville property

KW - Non-commutative peaking

KW - Rigidity

KW - Stochastic matrices

KW - Tensor algebras

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Doob equivalence and non-commutative peaking for Markov chains

Abstract

Bibliographical note

Keywords

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