Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

TL;DR

This work develops a noncommutative function-theoretic framework for algebras of bounded analytic nc functions on nc subvarieties of the nc unit ball. It identifies H^(rak V) with Mult(H^2_{rak V}) and proves a complete isometric identification H^(rak V)  H^(rak B_d)/rak J_{rak V}, generalizing quotient descriptions known in the commutative case. The authors establish a comprehensive isomorphism theory: completely isometric isomorphisms between such algebras are implemented by nc biholomorphisms, with stronger homogeneous-case results yielding conformal equivalence via automorphisms and unitary maps. A homogeneous nc Nullstellensatz is proved, and the study extends to algebras of continuous functions, showing A(rak V) = rak A_{rak V} for homogeneous varieties and providing a parallel classification. The work also connects these nc-analytic algebras to subproduct systems and clarifies links to the commutative case, including a version of the commutative free Nullstellensatz. Overall, the paper delivers a cohesive nc-analytic treatment of function algebras on variegated nc domains, with sharp isomorphism/classification results and deep ties to operator-algebraic structures.

Abstract

We study algebras of bounded, noncommutative (nc) analytic functions on nc subvarieties of the nc unit ball. Given a nc variety $\mathfrak{V}$ in the nc unit ball $\mathfrak{B}_d$, we identify the algebra of bounded analytic functions on $\mathfrak{V}$ --- denoted $H^\infty(\mathfrak{V})$ --- as the multiplier algebra $\operatorname{Mult} \mathcal{H}_{\mathfrak{V}}$ of a certain reproducing kernel Hilbert space $\mathcal{H}_{\mathfrak{V}}$ consisting of nc functions on $\mathfrak{V}$. We find that every such algebra $H^\infty(\mathfrak{V})$ is completely isometrically isomorphic to the quotient $H^\infty(\mathfrak{B}_d)/ \mathcal{J}_{\mathfrak{V}}$ of the algebra of bounded nc holomorphic functions on the ball by the ideal $\mathcal{J}_{\mathfrak{V}}$ of bounded nc holomorphic functions which vanish on $\mathfrak{V}$. We investigate the problem of when two algebras $H^\infty(\mathfrak{V})$ and $H^\infty(\mathfrak{W})$ are isometrically isomorphic. If the variety $\mathfrak{W}$ is the image of $\mathfrak{V}$ under a nc analytic automorphism of $\mathfrak{B}_d$, then $H^\infty(\mathfrak{V})$ and $H^\infty(\mathfrak{W})$ are (completely) isometrically isometric. We prove that the converse holds in the case where the varieties are homogeneous; in general we can only show that if the algebras are isometrically isomorphic, then there must be nc holomorphic maps between the varieties. Along the way we are led to consider some interesting problems on function theory in the nc unit ball. For example, we study various versions of the Nullstellensatz (that is, the problem of to what extent an ideal is determined by its zero set), and we obtain perfect Nullstellensatz in both the homogeneous as well as the commutative cases. We also consider similar problems regarding the bounded analytic functions that extend continuously to the boundary of $\mathfrak{B}_d$.

Theorems & Definitions (156)

• Theorem : Theorem \ref{['thm:isomorphism']}
• Theorem : Theorem \ref{['thm:null_poly']}
• Theorem : Theorem \ref{['thm:isomorphism_homo']} and Corollary \ref{['cor:equivalence_is_linear_homog']}
• Corollary : Commutative free Nullstellensatz --- Corollary \ref{['cor:free_com_NSTZ']}
• Remark 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof