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. 1 1− z,w (see [11] and [24]). This space is a complete Pick space, i.e., the multipliers of the Drury-Arveson space admit an interpolation theorem for matrix valued functions generalizing the classical Nevanlina-Pick interpolation theorem in the unit disc. Let us write M d for the algebra of multipliers on H 2 d , it is a maximal abelian WOT-closed operator subalgebra of B(H 2 d ) generated by the operators M z j of multiplication by coordinate functions. In [3, Theorem 8.2] it was shown that the Drury-Arveson space for d = ∞ is the universal complete Pick space, namely, if H is a separable complete Pick reproducing kernel Hilbert space on a set X with kernel k, then there exists an embedding b : X → B ∞ and a nowhere vanishing function δ on X, such that k(x, y) = δ(x)δ(y)k ∞ (b(x), b(y)) and H is isometrically embedded in δH 2 d . Let V ⊂ B d be an analytic subvariety of B d cut out by functions in M d . We can associate to it a reproducing kernel Hilbert space H V spanned by kernel functions k d (·, w) for w ∈ V . This space turns out to be a complete Pick space and the multiplier algebraof functions vanishing on V . It is thus natural to ask to what extent does the algebra M V determine the variety V and vice versa. The isomorphism problem for subvarieties of B d cut out by multipliers of the Drury-Arveson space was studied by Davidson, Ramsey and Shalit. In [22] and [23] they studied the algebra M V and its norm closed analog and proved that if V, W ⊂ B d are subvarieties of B d , such that their affine span is all of C d , then M V is completely isometrically isomorphic to M W if and only if there is an automorphism of B d mapping V onto W (see also [64] for a survey and more results on the commutative isomorphism problem). One of the main tools in the proof of the theorem is a theorem that appears both in [63] and [39] and states that the fixed point set of a self map of B d is the arXiv:1709.00446v2 [math.OA] 24 Dec 2018 intersection of B d with an affine subspace (see also [32, Theorem 23.2] and [42, Theorem 6.3]). The noncommutative (nc for short), or free functions were introduced by Taylor in [66] and [67]. Taylor's goal was to facilitate noncommutative functional calculus and thus he endeavored to give topological algebras analogous to the classical Frechet algebras of analytic functions on open domains in C d . Voiculescu in [72], [73], [74] and [75] developed the ideas of Taylor in the context of free probability. Helton, Klep , McCullough and Schweighofer applied noncommutative analysis in order to obtain dimension free relaxation of the LMI containment problem (see [34] and [35]). Their results were extended and impr...