Holomorphic functions on the quantum polydisk and on the quantum ball

TL;DR

This paper develops noncommutative analogs of holomorphic function algebras on polydisks and balls, exploring their properties, relationships, and non-isomorphism under certain conditions, extending classical complex analysis results into the quantum setting.

Contribution

It introduces and analyzes quantum versions of holomorphic function algebras on polydisks and balls, establishing their properties and non-isomorphism for specific parameters.

Findings

Constructed quantum holomorphic function algebras for all nonzero q

Established relation between quantum ball algebra and Vaksman's algebra for 0<q<1

Proved non-isomorphism of quantum polydisk and ball algebras when |q|=1 and n≥2

Abstract

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].