Shamovich, Eli2018-10-182018-10-182018-07-01https://dx.doi.org/10.1016/j.jfa.2018.03.004http://hdl.handle.net/10012/14017The final publication is available at Elsevier via https://dx.doi.org/10.1016/j.jfa.2018.03.004 © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/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.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Noncommutative functionsOperator algebrasComplete Pick spacesFree ballArticle