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

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 $\mathfrak{B}_d$.