Holomorphically finitely generated algebras

TL;DR

This paper introduces holomorphically finitely generated (HFG) Fréchet algebras as analytic analogs of finitely generated algebraic structures, establishing their categorical equivalence with certain Stein spaces and providing concrete examples including quantum and free structures.

Contribution

It defines HFG algebras, proves their categorical equivalence with Stein spaces, and constructs various examples including quantum and free analytic spaces.

Findings

HFG algebras are anti-equivalent to Stein spaces of finite embedding dimension.

The class of HFG algebras is stable under natural algebraic constructions.

Examples include holomorphic functions on quantum and free spaces.

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 natural 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 polyannulus.