Complex Analysis and Operator Theory

, Volume 6, Issue 2, pp 465–475 | Cite as

Projective Freeness of Algebras of Real Symmetric Functions

  • Amol SasaneEmail author


Let \({\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}\), and let \({\overline{\mathbb{D}}^n}\) denote its closure in \({\mathbb {C}^n}\). Consider the ring
$$C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C}) =\left\{f: \overline{\mathbb{D}}^n\rightarrow \mathbb {C}:f \,\, {\rm is \,\, continuous \,\, and}\,\, f(z)=\overline{f(\overline{z})} \;(z\in \overline{\mathbb{D}}^n)\right\}$$
with pointwise operations, where \({\overline{\cdot}}\) is used appropriately to denote both (componentwise) complex conjugation and closure. It is shown that \({C_{\rm r}(\overline{\mathbb{D}}^n;\mathbb {C})}\) is projective free, that is, finitely generated projective modules over \({C_{\rm r}(\overline{\mathbb{D}}^n; \mathbb {C})}\) are free. Let A denote the polydisc algebra, that is, the set of all continuous functions defined on \({\overline{\mathbb{D}}^n}\) that are holomorphic in \({\mathbb {D}^n}\). For N a positive integer, let ∂N A denote the algebra of functions \({f \in A}\) whose complex partial derivatives of all orders up to N belong to A. We show the projective freeness of each of the real symmetric algebras \({\partial^{-N}A_{\rm r}=\{{f \in \partial^{-N} A: f(z)=\overline{f(\overline{z})}\;(z\in \overline{\mathbb{D}}^n)}\}}\).


Real Banach algebras Projective free rings Serre’s conjecture Real symmetric function algebras Control theory 

Mathematics Subject Classification (2000)

Primary 46J40 Secondary 13C10 46H20 46M10 54C40 93D15 


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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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