Holomorphically finitely generated algebras

Abstract

We introduce and study holomorphically finitely generated (HFG) Fréchet algebras, which are analytic counterparts of affine (i.e., finitely generated) 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.

Quantum polydisk, quantum ball, andq-analog of Poincaré's theorem

“…The explicit construction (1) of O q (C n ) leads naturally to the following definition. 35,36]). Let q ∈ C × , and let r > 0.…”

“…Recall from [36] (see also [35]) that each family (A i ) i∈I of Arens-Michael algebras has a coproduct * i∈I A i in the category AM of Arens-Michael algebras. The algebra * i∈I A i is called the Arens-Michael free product of the A i 's.…”

“…The algebra * i∈I A i is called the Arens-Michael free product of the A i 's. By definition [35,36], the algebra of holomorphic functions on the free n-polydisk of radius r ∈ (0, +∞] is F (D n r ) = O(D r ) * · · · * O(D r ).…”

“…Moreover, for each ρ ∈ (0, r) the norm · ρ on O q (D n r ) given by (2) is equal to the quotient of the norm · ρ on F T (D n r ) given by (8). Part (i) of Theorem 6.1, except for the equality of the norms, was proved in [36,Theorem 7.13] in the more general multiparameter case. Part (ii) is new.…”

“…For each q ∈ C × we define q-analogs of the algebras O(D n ) and O(B n ) to be the completions of the algebraic quantum affine space O reg q (C n ) with respect to certain families of seminorms. Both the resulting algebras O q (D n ) and O q (B n ) were introduced earlier in [35] (see also [36] for a more detailed treatment of O q (D n )), but the definition of O q (B n ) was given only for 0 < q < 1. Here we propose a different approach to O q (B n ) which makes sense for all q ∈ C × and which is equivalent to the approach of [35] in the case where 0 < q < 1.…”