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Projective Freeness of Algebras of Real Symmetric Functions
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2012 (English)In: Complex Analysis and Operator Theory, ISSN 1661-8254, E-ISSN 1661-8262, Vol. 6, no 2, p. 465-475Article in journal (Refereed) Published
##### Abstract [en]

Let D-n := {z = (z(1),...,z(n)) is an element of C-n : vertical bar z(j)vertical bar < 1, j = 1,...,n}, and let <(D)over bar>(n) denote its closure in C-n. Consider the ring C-r((D) over bar (n); C) = {f : (D) over bar (n) -> C : f is continuous and f (z) = <(f<(z)over bar>)over bar> (z is an element of (D) over bar (n))} with pointwise operations, where u is used appropriately to denote both (componentwise) complex conjugation and closure. It is shown that C-r((D) over bar (n); C) is projective free, that is, finitely generated projective modules over C-r((D) over bar (n); C) are free. Let A denote the polydisc algebra, that is, the set of all continuous functions defined on (D) over bar (n) that are holomorphic in D-n. For N a positive integer, let partial derivative(-N) A denote the algebra of functions f is an element of 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 derivative(-N) A(r) = {f is an element of partial derivative(-N) A : f (z) = <(f<(z)over bar>)over bar> (z is an element of (D) over bar (n))}.

##### Place, publisher, year, edition, pages
2012. Vol. 6, no 2, p. 465-475
##### Keyword [en]
Real Banach algebras, Projective free rings, Serre's conjecture, Real symmetric function algebras, Control theory
Mathematics
##### Identifiers
ISI: 000301976500009Scopus ID: 2-s2.0-84858792109OAI: oai:DiVA.org:kth-93916DiVA, id: diva2:525142
##### Note
QC 20120507Available from: 2012-05-07 Created: 2012-05-03 Last updated: 2017-12-07Bibliographically approved

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Cite
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