Doob equivalence and non-commutative peaking for Markov chains
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Doob equivalence and non-commutative peaking for Markov chains. / Chen, Xinxin; Dor-On, Adam; Hui, Langwen; Linden, Christopher; Zhang, Yifan.
In: Journal of Noncommutative Geometry, Vol. 15, No. 4, 2021, p. 1469-1484.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Doob equivalence and non-commutative peaking for Markov chains
AU - Chen, Xinxin
AU - Dor-On, Adam
AU - Hui, Langwen
AU - Linden, Christopher
AU - Zhang, Yifan
N1 - Publisher Copyright: © 2021 European Mathematical Society Published by EMS Press This work is licensed under a CC BY 4.0 license
PY - 2021
Y1 - 2021
N2 - 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.
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.
KW - Doob equivalence
KW - Harmonic functions
KW - Liouville property
KW - Non-commutative peaking
KW - Rigidity
KW - Stochastic matrices
KW - Tensor algebras
U2 - 10.4171/JNCG/444
DO - 10.4171/JNCG/444
M3 - Journal article
AN - SCOPUS:85123782504
VL - 15
SP - 1469
EP - 1484
JO - Journal of Noncommutative Geometry
JF - Journal of Noncommutative Geometry
SN - 1661-6952
IS - 4
ER -
ID: 297058510