**Title: **Ergodic Quantum Processes on Finite von Neumann Algebras

**Abstract: ** Let $(M,\tau)$ be a tracial von Neumann algebra with a separable predual and let $(\Omega, \mathbb{P})$ be a probability space. An ergodic quantum process on $M$ is a composition of the form $\gamma_{T^n\omega}\circ \gamma_{T^{n-1}\omega}\circ \cdots \circ \gamma_{T^m\omega}$, where $\gamma\colon \Omega \times L^1(M,\tau)\to L^1(M,\tau)$ is a bounded positive linerar operator, $T\in \text{Aut}(\Omega, \mathbb{P})$ is ergodic, and $n,m\in \mathbb{Z}$. Physically, such processes model a discrete time evolution of a quantum system that is subject to (ergodically constrained) disorder. Movassagh and Schenker recently studied ergodic quantum processes in the finite dimensional case $M=M_n(\mathbb{C})$, and they showed that under reasonable assumptions such processes collapse to rank-one maps on $L^1(M,\tau)$ exponentially fast almost surely. In this talk, I will discuss how to generalize their results to all separable finite von Neumann algebras. Essential to the analysis in the infinite dimensional case is the so-called Hennion metric on the normal state space of $M$, which is defined using the natural ordering on $L^1(M,\tau)_+$.

This is based on joint work with Eric B. Roon.