Toeplitz quotient $C^*$-algebras and ratio limits for random walks

Abstract

We study quotients of the Toeplitz C * -algebra of a random walk, similar to those studied by the author and Markiewicz for finite stochastic matrices. We introduce a new Cuntz-type quotient C * -algebra for random walks that have convergent ratios of transition probabilities. These C * -algebras give rise to new notions of ratio limit space and boundary for such random walks, which are computed by appealing to a companion paper by Woess. Our combined results are leveraged to identify a unique symmetry-equivariant quotient C *algebra for any symmetric random walk on a hyperbolic group, shedding light on a question of Viselter on C * -algebras of subproduct systems.

Ratio limits and Martin boundary

“…In relation with his current work on Cuntz algebras related to random walks, Adam Dor-On, in an exchange that lead to the "companion" paper [11], has asked how one can describe this compactification in terms of a given structure of the underlying state space, and how it relates with the ρ -Martin compactification. His questions were motivated by operator algebraic issues, and the answers provided here contribute to clarifying some of them.…”

“…The companion paper [11] makes crucial use of the following variant of the ratio limit compactification. Let ∼ be the equivalence relation on X such that…”

“…is a subgroup of Γ, see [11]. Since the ratio limit kernel h(x, y) in (1.1) satisfies h(x, y) = f (x −1 y), where f (x) = h(x, e), one gets that…”

“…[11] calls this the (ordinary) ratio limit compactification.Documenta Mathematica 26 (2021) 1501-1528…”

Ratio-limit boundaries of random walks on relatively hyperbolic groups

“…Thus, completely new techniques are often necessary in order to prove the existence of a natural co-universal quotient in various scenarios (see for instance [3,42]). By using deep results from the theory of random walks [38,68], the first author established the existence of a natural co-universal quotient for Toeplitz C*-algebras arising from random walks, when the random walks are symmetric, aperiodic, and on non-elementary hyperbolic groups [21,Corollary 5.2].…”