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

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 $\mathrm{Mult}{\mathcal{H}}_{\mathfrak{V}}$ 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 ${\mathfrak{B}}_d$.

52 pages. Final version. To appear in Trans. Amer. Math. Soc

References in corpus (5)

• The isomorphism problem for some universal operator algebras
• Free holomorphic automorphisms of the unit ball of $B(H{)}^n$
• Free Convex Algebraic Geometry
• CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond
• Free holomorphic functions on the unit ball of B(H)^n

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• Bianalytic free maps between spectrahedra and spectraballs
• On fixed points of self maps of the free ball
• Noncommutative partial convexity via $\Gamma$-convexity
• On the classification of function algebras on subvarieties of noncommutative operator balls
• Factorization of noncommutative polynomials and Nullstellensätze for the free algebra
• A polynomial approximation result for free Herglotz-Agler functions
• Tensor algebras of subproduct systems and noncommutative function theory
• Plurisubharmonic Noncommutative Rational Functions
• Noncommutative partially convex rational functions
• Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball
• Weak-* and completely isometric structure of noncommutative function algebras
• The Arveson boundary of a Free Quadrilateral is given by a noncommutative variety