Quantum polydisk, quantum ball, and a q-analog of Poincar\'e's theorem

The classical Poincar\'e theorem (1907) asserts that the polydisk $\mathbb D^n$ and the ball $\mathbb B^n$ in $\mathbb C^n$ are not biholomorphically equivalent for $n\ge 2$. Equivalently, this means that the Fr\'echet algebras $\mathcal O(\mathbb D^n)$ and $\mathcal O(\mathbb B^n)$ of holomorphic functions are not topologically isomorphic. Our goal is to prove a noncommutative version of the above result. Given $q\in\mathbb C\setminus\{ 0\}$, we define two noncommutative power series algebras $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$, which can be viewed as $q$-analogs of $\mathcal O(\mathbb D^n)$ and $\mathcal O(\mathbb B^n)$, respectively. Both $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ are the completions of the algebraic quantum affine space $\mathcal O_q^{\mathrm{reg}}(\mathbb C^n)$ w.r.t. certain families of seminorms. In the case where $0<q<1$, the algebra $\mathcal O_q(\mathbb B^n)$ admits an equivalent definition related to L. L. Vaksman's algebra of continuous functions on the closed quantum ball. We show that both $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ can be interpreted as Fr\'echet algebra deformations (in a suitable sense) of $\mathcal O(\mathbb D^n)$ and $\mathcal O(\mathbb B^n)$, respectively. Our main result is that $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ are not isomorphic if $n\ge 2$ and $|q|=1$, but are isomorphic if $|q|\ne 1$.