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.

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非对易球子变体上的非对易函数代数：有界和完全有界同构问题

给定 nc 单位球 $\mathfrak{B}_d$ 中的非交换 (nc) 变体 $\mathfrak{V}$，我们考虑有界 nc 全纯函数的代数 $H^\infty(\mathfrak{V})$在 $\mathfrak{V}$ 上。我们研究了当两个代数 $H^\infty(\mathfrak{V})$ 和 $H^\infty(\mathfrak{W})$ 同构时的问题。我们证明这些代数是弱 $*$ 连续同构的当且仅当在相似性包络之间存在 nc 双全同态 $G : \widetilde{\mathfrak{W}} \to \widetilde{\mathfrak{V}}$这是关于自由伪双曲度量的双利普希茨。此外，这样的同构总是具有 $f \mapsto f \circ G$ 的形式，其中 $G$ 是 nc 双全同构。这些结果也为 Davidson--Pitts 和 Popescu 研究的非交换解析 Toeplitz 代数 $H^\infty(\mathfrak{B}_d)$ 的自同构提供了一些新的启示。特别地，我们发现 $\operatorname{Aut}(H^\infty(\mathfrak{B}_d))$ 是 $\operatorname{Aut}(\widetilde{\mathfrak{B}}_d) 的真子群$. 当 $d<\infty$ 并且变体是齐次时，我们去除了弱 $*$ 连续性假设，表明两个这样的代数是有界同构的当且仅当在nc 品种。我们提供两个证明。在非对易设置中，我们的主要工具是非对易谱半径，我们证明了几个新结果。在自由交换情况下，