Banach algebras of symmetric functions on the polydisc

Abstract: Let ${\mathbb{D}}={z\in \mathbb{C}:|z|&lt;1}$ and for an integer $d\geq 1$, let $S_d$ denote the symmetric group, consisting of of all permutations of the set ${1,\cdots, d}$. A function $f:{\mathbb{D}}^d\rightarrow \mathbb{C}$ is symmetric if $$f(z_1,\cdots, z_d)=f(z_{\sigma(1)},\cdots, z_{\sigma (d)})$$ for all $\sigma \in S_d$ and all $(z_1,\cdots, z_d)\in {\mathbb{D}}<sup>d$. The polydisc algebra $A({\mathbb{D}}<sup>d)$ is the Banach algebra of all holomorphic functions $f$ on the polydisc ${\mathbb{D}}<sup>d$ that can be continuously extended to the closure of the polydisc in ${\mathbb{C}}<sup>d$, with pointwise operations and the supremum norm (given by $$|f|<em>\infty:=\sup</em>{\mathbf{z} \in {\mathbb{D}}^d} |f(\mathbf{z})|$$). Let $A_{\textrm{sym}}({\mathbb{D}}<sup>d)$ be the Banach subalgebra of $A({\mathbb{D}}<sup>d)$ consisting of all symmetric functions in the polydisc algebra. Algebraic-analytic properties of $A_{\textrm{sym}}({\mathbb{D}}<sup>d)$ are investigated. In particular, the following results are shown: the corona theorem, description of the maximal ideal space and its contractibility, Hermiteness, projective-freeness, and non-coherence.