Projective Freeness of Algebras of Real Symmetric Functions

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))}.