Algebras of noncommutative functions on subvarieties of the noncommutative ball: the bounded and completely bounded isomorphism problem
Guy Salomon, Orr Shalit, Eli Shamovich
TL;DR
This work classifies algebras of bounded nc holomorphic functions on nc varieties inside the nc ball by geometric data encoded in the similarity envelopes. The authors introduce the similarity envelope and two free pseudo-hyperbolic distances, proving that weak-$*$, bounded, and cb isomorphisms between $H^{\infty}({\mathfrak V})$ and $H^{\infty}({\mathfrak W})$ are equivalent to the existence of a bi-Lipschitz nc biholomorphism between $\widetilde{{\mathfrak V}}$ and $\widetilde{{\mathfrak W}}$, with the isomorphism acting via precomposition with this map. In the homogeneous finite-dimensional case, they remove the weak-$*$ continuity assumption and obtain sharp equivalences with bi-Lipschitz linear maps on the similarity envelopes, using a noncommutative spectral radius and a noncommutative Schwarz lemma to underpin the analysis. The paper also develops a commutative nc Nullstellensatz, extends the theory to the algebras $A({\mathfrak V})$, and discusses concrete examples such as $q$-commutation varieties, yielding a robust framework for understanding automorphisms and isomorphisms of nc multiplier algebras through geometric and spectral data. The results bridge operator algebra structure with nc-analytic geometry, offering tools for classifying algebras by bi-Lipschitz nc biholomorphisms of their similarity envelopes and illuminating the role of the joint spectral radius in noncommutative isomorphism problems.
Abstract
Given a noncommutative (nc) variety $\mathfrak{V}$ in the nc unit ball $\mathfrak{B}_d$, we consider the algebra $H^\infty(\mathfrak{V})$ of bounded nc holomorphic functions on $\mathfrak{V}$. We investigate the problem of when two algebras $H^\infty(\mathfrak{V})$ and $H^\infty(\mathfrak{W})$ are isomorphic. We prove that these algebras are weak-$*$ continuously isomorphic if and only if there is an nc biholomorphism $G : \widetilde{\mathfrak{W}} \to \widetilde{\mathfrak{V}}$ between the similarity envelopes that is bi-Lipschitz with respect to the free pseudo-hyperbolic metric. Moreover, such an isomorphism always has the form $f \mapsto f \circ G$, where $G$ is an nc biholomorphism. These results also shed some new light on automorphisms of the noncommutative analytic Toeplitz algebras $H^\infty(\mathfrak{B}_d)$ studied by Davidson--Pitts and by Popescu. In particular, we find that $\operatorname{Aut}(H^\infty(\mathfrak{B}_d))$ is a proper subgroup of $\operatorname{Aut}(\widetilde{\mathfrak{B}}_d)$. When $d<\infty$ and the varieties are homogeneous, we remove the weak-$*$ continuity assumption, showing that two such algebras are boundedly isomorphic if and only if there is a bi-Lipschitz nc biholomorphism between the similarity envelopes of the nc varieties. We provide two proofs. In the noncommutative setting, our main tool is the noncommutative spectral radius, about which we prove several new results. In the free commutative case, we use a new free commutative Nullstellensatz that allows us to bootstrap techniques from the fully commutative case.