Noncommutative analogues of Stein spaces of finite embedding dimension

TL;DR

This paper introduces holomorphically finitely generated (HFG) Fréchet algebras as noncommutative analogues of Stein spaces, establishing their properties, examples, and deformation relationships with classical holomorphic function algebras.

Contribution

It defines HFG algebras, proves their categorical equivalence to Stein spaces, and provides concrete examples including quantum and free polydisk algebras.

Findings

HFG algebras are stable under standard constructions

Quantum polydisk and quantum ball algebras are isomorphic

HFG algebras can be viewed as deformations of classical holomorphic function algebras

Abstract

We introduce and study holomorphically finitely generated (HFG) Fr\'echet algebras, which are analytic counterparts of affine (i.e., finitely generated) $\mathbb C$-algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG algebras is stable under some standard constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions on the quantum affine space and on the quantum torus), the algebras of holomorphic functions on the free polydisk, on the quantum polydisk, and on the quantum ball. We further concentrate on the algebras of holomorphic functions on the quantum polydisk and on the quantum ball and show that they are isomorphic, in contrast to the classical case. Finally, we interpret our algebras as Fr\'echet algebra deformations of the classical algebras of holomorphic functions on the polydisk and on the ball in $\mathbb C^n$.