Banach algebras of symmetric functions on the polydisc

TL;DR

This paper studies the algebraic and analytic properties of symmetric functions on the polydisc, including the corona theorem, maximal ideal space, and structural properties of the symmetric polydisc algebra.

Contribution

It provides new results on the structure and properties of the symmetric polydisc algebra, including the corona theorem and maximal ideal space analysis.

Findings

Proved the corona theorem for the symmetric polydisc algebra.

Described the maximal ideal space and proved its contractibility.

Established properties like Hermiteness, projective-freeness, and non-coherence.

Abstract

Let ${\mathbb{D}}=\{z\in \mathbb{C}:|z|<1\}$ 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 $(z_1,\cdots, z_d)\in {\mathbb{D}}^d$. The polydisc algebra $A({\mathbb{D}}^d)$ is the Banach algebra of all holomorphic functions $f$ on the polydisc ${\mathbb{D}}^d$ that can be continuously extended to the closure of the polydisc in ${\mathbb{C}}^d$, with pointwise operations and the supremum norm (given by $\|f\|_\infty:=\sup_{\mathbf{z} \in {\mathbb{D}}^d} |f(\mathbf{z})|$). Let $A_{\textrm{sym}}({\mathbb{D}}^d)$ be the Banach subalgebra of $A({\mathbb{D}}^d)$ consisting of all symmetric functions in the polydisc algebra. Algebraic-analytic properties of $A_{\textrm{sym}}({\mathbb{D}}^d)$ 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.