Toeplitz quotient $C^{\ast }$-algebras and ratio limits for random walks

Abstract

We study quotients of the Toeplitz $C^{\ast }$-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 $C^{\ast }$-algebra for random walks that have convergent ratios of transition probabilities. These $C^{\ast }$-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 $C^{\ast }$-algebra for any symmetric random walk on a hyperbolic group, shedding light on a question of Viselter on $C^{\ast }$-algebras of subproduct systems.