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. 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...

Noncommutative Christoffel-Darboux kernels

“…We see that the fixed point set of f n is precisely the set of fixed points of ∆f (0 n , 0 n ) intersected with B d (n). Now with these theorems at hand, we can prove the following theorem, which is a strengthening of both [37,Theorem 4.1] and [19,Theorem 4.5]. Recall that for S ⊂ M d , the matrix span of S is mat-span(S) = ⊔ n∈N mat-span(S)(n), where…”

“…In particular, Popescu proved a noncommutative version of Wolff's theorem for the row ball [32,Theorem 3.1]. A noncommutative version of a fixed point theorem of Rudin [34] and Hervé [23] was obtained by the second author in [37]. The primary motivation for the proof of the latter fixed point theorem was the classification of quotients of the free semigroup algebra by WOT closed ideals up to completely isometric isomorphism [35,36].…”

“…Section 5 contains an nc version of the classical Vigué and Bedford theorems on iterations of holomorphic functions (Theorems 5.1 and 5.4). We use these results to obtain an extension of [37,Theorem 4.1] that is the full nc version of a result of Davidson, Ramsey, and Shalit [19,Theorem 4.4].…”

Algebras of noncommutative functions on subvarieties of the noncommutative ball: The bounded and completely bounded isomorphism problem

“…Let V ⊆ B d be a homogeneous nc variety containing only commuting d-tuples and let V = V(1) be the scalar level of V. Then there exists an integer 1 In [43], we also examined the nonhomogeneous case, and we showed that these algebras are completely isometrically isomorphic if and only if the varieties V and W are nc biholomorphic. The main result of [45] then shows that, at least when the varieties contain a scalar point, such an nc biholomorphism is just a restriction of an nc automorphism of the nc ball. 3 4 corresponding affine variety Z(I) = Z C d (I) = {z ∈ C d : p(z) = 0 for all p ∈ I}.Together with this geometric object, there are two natural algebraic objects: the quotient C[z]/I -which is the universal unital algebra generated by d commuting elements satisfying the relations in I -and the algebra of regular functions:Consider two ideals I, J C[z].…”

“…In[43], we also examined the nonhomogeneous case, and we showed that these algebras are completely isometrically isomorphic if and only if the varieties V and W are nc biholomorphic. The main result of[45] then shows that, at least when the varieties contain a scalar point, such an nc biholomorphism is just a restriction of an nc automorphism of the nc ball.…”