Holomorphic functions on the quantum polydisk and on the quantum ball

We introduce and study noncommutative (or ``quantized'') versions of the algebras of holomorphic functions on the polydisk and on the ball in $\mathbb C^n$. Specifically, for each $q\in\mathbb C\setminus\{ 0\}$ we construct Fr\'echet algebras $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ such that for $q=1$ they are isomorphic to the algebras of holomorphic functions on the open polydisk $\mathbb D^n$ and on the open ball $\mathbb B^n$, respectively. In the case where $0<q<1$, we establish a relation between our holomorphic quantum ball algebra $\mathcal O_q(\mathbb B^n)$ and L. L. Vaksman's algebra $C_q(\bar{\mathbb B}^n)$ of continuous functions on the closed quantum ball. Finally, we show that $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$ are not isomorphic provided that $|q|=1$ and $n\ge 2$. This result can be interpreted as a $q$-analog of Poincar\'e's theorem, which asserts that $\mathbb D^n$ and $\mathbb B^n$ are not biholomorphically equivalent unless $n=1$. This paper replaces the first part of Version 1: arXiv:1508.05768v1 [math.FA].