**BCRI Mini-Symposium**: **Noncommutative Probability & Quantum Information**

**Monday, 10**^{th} October 2022 from 12:00 to 15:00

**Organizers: Claus Koestler (UCC), Stephen Wills (UCC)**

**SPEAKER: **J.P. McCarthy (Munster Technological University)

**TITLE:** The Kawada-Itô theorem for finite quantum groups.

**ABSTRACT:** *Necessary and sufficient conditions for a Markov chain to be ergodic are that the chain is irreducible and aperiodic. This result is manifest in the case of random walks on finite groups by a statement about the support of the driving probability: a random walk on a finite group is ergodic if and only if the support is not concentrated on a proper subgroup, nor on a coset of a proper normal subgroup. The study of random walks on finite groups extends naturally to the study of random walks on compact quantum groups, where a state on the algebra of functions plays the role of the driving probability. A random walk on a compact quantum group can fail to be irreducible without being concentrated on a proper quantum subgroup. In this talk we will explore this phenomenon. Time allowing, we will talk about periodicity, and as a conclusion, I give necessary and sufficient conditions for ergodicity of a random walk on a finite quantum group in terms of the support projection of the driving state.*

In the end the talk (below) didn’t quite match the abstract.

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