Iteration theory of noncommutative maps

TL;DR

This paper develops an iteration theory for noncommutative self-maps on matrix convex domains, extending classical theorems like Denjoy-Wolff and Wolff's to noncommutative settings and quotient algebras.

Contribution

It proves noncommutative versions of the Denjoy-Wolff and Wolff theorems and generalizes fixed point results to quotients of free semigroup algebras.

Findings

Established a noncommutative Denjoy-Wolff theorem for the row ball.

Proved a Wolff-type theorem for general matrix convex sets.

Extended fixed point results to quotients of free semigroup algebras.

Abstract

This note aims to study the iteration theory of noncommutative self-maps of bounded matrix convex domains. We prove a version of the Denjoy-Wolff theorem for the row ball and the maximal quantization of the unit ball of $\mathbb{C}^d$. For more general bounded matrix convex sets, we prove a version of Wolff's theorem inspired by the results of Abate. Lastly, we use iteration and fixed point theory to generalize the commutative results of Davidson, Ramsey, and Shalit to quotients of the free semigroup algebra by WOT closed ideals.