  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.

Operator Algebras