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 an 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 $\mathfrak {V}$. In order to demonstrate this isomorphism, we prove that the space $\mathcal {H}_{\mathfrak {V}}$ is an nc complete Pick space (a fact recently proved—by other methods—by Ball, Marx, and Vinnikov).

We investigate the problem of when two algebras $H^\infty (\mathfrak {V})$ and $H^\infty (\mathfrak {W})$ are (completely) isometrically isomorphic. If the variety $\mathfrak {W}$ is the image of $\mathfrak {V}$ under an nc analytic automorphism of $\mathfrak {B}_d$, then $H^\infty (\mathfrak {V})$ and $H^\infty (\mathfrak {W})$ are completely isometrically isomorphic. 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 completely isometrically isomorphic, then there must be nc holomorphic maps between the varieties (in the case $d = \infty$ we need to assume that the isomorphism is also weak-$*$ continuous).

We also consider similar problems regarding the bounded analytic functions that extend continuously to the boundary of $\mathfrak {B}_d$ and related norm closed algebras; the results in the norm closed setting are somewhat simpler and work for the case $d = \infty$ without further assumptions.

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.