On fixed points of self maps of the free ball

TL;DR

This paper investigates the structure of fixed points of noncommutative self maps of the free ball, revealing they form intersections with linear subspaces, and applies findings to isomorphism problems in noncommutative function theory.

Contribution

It characterizes fixed point sets of noncommutative self maps fixing the origin and applies results to multiplier algebra isomorphism problems.

Findings

Fixed point sets are intersections with linear subspaces

Fixed points are characterized on every level of the free ball

Application to isomorphism problems in noncommutative spaces

Abstract

In this paper, we study the structure of the fixed point sets of noncommutative self maps of the free ball. We show that for such a map that fixes the origin the fixed point set on every level is the intersection of the ball with a linear subspace. We provide an application for the completely isometric isomorphism problem of multiplier algebras of noncommutative complete Pick spaces.