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Space-time boundaries for random walks and their application to operator algebras

Adam Dor-On, Matthieu Dussaule, Ilya Gekhtman, Pavel Prudnikov

Abstract

We investigate the Martin boundary of the space-time Markov chain associated to a finitely supported random walk $(Γ, μ)$ with spectral radius $ρ$ and relate it to several classical compactifications of $Γ$. Assuming the strong ratio-limit property, we prove that the reduced ratio-limit compactification embeds naturally into the space-time Martin boundary. We introduce the $0$-Martin boundary, which governs the behaviour of $\infty$-harmonic functions, and show that the $0$-Martin kernels arise as rescaled limits of $λ$-Martin kernels as $λ\rightarrow 0$. For symmetric random walks on hyperbolic groups, the $0$-Martin boundary naturally covers the Gromov boundary, while the cover need not be injective in general. Our main structural theorem identifies the minimal space-time Martin boundary with the disjoint union of minimal $λ$-Martin boundaries over $λ\in [0, ρ^{-1}]$ with its natural pointwise topology. As an application, we show that the noncommutative Shilov boundary of the tensor algebra of the random walk $(Γ, μ)$ coincides with its Toeplitz $C^*$-algebra.

Space-time boundaries for random walks and their application to operator algebras

Abstract

We investigate the Martin boundary of the space-time Markov chain associated to a finitely supported random walk with spectral radius and relate it to several classical compactifications of . Assuming the strong ratio-limit property, we prove that the reduced ratio-limit compactification embeds naturally into the space-time Martin boundary. We introduce the -Martin boundary, which governs the behaviour of -harmonic functions, and show that the -Martin kernels arise as rescaled limits of -Martin kernels as . For symmetric random walks on hyperbolic groups, the -Martin boundary naturally covers the Gromov boundary, while the cover need not be injective in general. Our main structural theorem identifies the minimal space-time Martin boundary with the disjoint union of minimal -Martin boundaries over with its natural pointwise topology. As an application, we show that the noncommutative Shilov boundary of the tensor algebra of the random walk coincides with its Toeplitz -algebra.
Paper Structure (9 sections, 29 theorems, 148 equations)

This paper contains 9 sections, 29 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Harmonic functions and $\lambda$-Martin boundaries.
  4. Operator algebras and representations.
  5. Space-time and Ratio-limit boundaries
  6. The 0-Martin boundary
  7. 0-Martin boundaries of hyperbolic groups
  8. Minimal space-time boundary
  9. C*-envelopes of tensor algebras arising from random walks

Key Result

Proposition 2.2

The map $\xi\in \partial_{M, \lambda}\Gamma\mapsto K(\cdot, \xi \,|\, \lambda)\in\mathcal{B}^{+}_{1}(\Gamma, {\mathbb{R}})$ is one-to-one and continuous. Moreover, if $\mu$ is finitely supported, then the map $y\in \Delta_{M, \lambda}\Gamma\mapsto K(\cdot, y \,|\, \lambda)$ is also one-to-one and co

Theorems & Definitions (69)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 2.4
  • Theorem 2.5: Martin-Poisson representation theorem
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 59 more