Journal of Noncommutative Geometry

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Volume 9, Issue 1, 2015, pp. 215–264
DOI: 10.4171/JNCG/192

Published online: 2015-04-13

Holomorphically finitely generated algebras

Alexei Yu. Pirkovskii[1]

(1) Higher School of Economics, Moscow, Russian Federation

We introduce and study holomorphically finitely generated (HFG) Fréchet 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.

Keywords: Fréchet algebra, HFG algebra, Stein space, Forster’s theorem, smash product, free polydisk, quantum polydisk

Pirkovskii Alexei: Holomorphically finitely generated algebras. J. Noncommut. Geom. 9 (2015), 215-264. doi: 10.4171/JNCG/192