Doob equivalence and non-commutative peaking for Markov chains

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

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Pages (from-to)1469-1484
Number of pages16
JournalJournal of Noncommutative Geometry
Issue number4
StatePublished - 2021
Externally publishedYes

Bibliographical note

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


  • 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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