On fixed points of self maps of the free ball

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.

References in corpus (7)

• Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball
• The isomorphism problem for some universal operator algebras
• Free holomorphic automorphisms of the unit ball of $B(H{)}^n$
• Minimal and maximal matrix convex sets
• Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization
• Free holomorphic functions on the unit ball of B(H)^n
• A Julia--Carathéodory theorem for hyperbolically monotone mappings in the Hilbert ball

Cited by in corpus (5)

• Algebras of noncommutative functions on subvarieties of the noncommutative ball: the bounded and completely bounded isomorphism problem
• On the classification of function algebras on subvarieties of noncommutative operator balls
• Tensor algebras of subproduct systems and noncommutative function theory
• Noncommutative hyperbolic metrics
• Weak-* and completely isometric structure of noncommutative function algebras