Projective freeness of algebras of real symmetric functions

TL;DR

This paper proves that the algebra of continuous real symmetric functions on the polydisk is projective free, meaning all finitely generated projective modules over it are free, and extends this result to certain subalgebras.

Contribution

It establishes the projective freeness of the algebra of real symmetric continuous functions on the polydisk, a new result in the theory of function algebras.

Findings

C_r is projective free.

Several subalgebras of the real symmetric polydisc algebra are also projective free.

Finitely generated projective modules over these algebras are free.

Abstract

Let D^n be the closed unit polydisk in C^n. Consider the ring C_r of complex-valued continuous functions on D^n that are real symmetric, that is, f(z)=(f(z^*))^* for all z in D^n. It is shown that C_r is projective free, that is, finitely generated projective modules over C_r are free. We also show that several subalgebras of the real symmetric polydisc algebra are projective free.