Embedding polydisk algebras into the disk algebra and an application to stable ranks

TL;DR

This paper demonstrates how to embed polydisk algebras into the disk algebra, enabling the construction of subalgebras with any desired stable rank, thus advancing the understanding of algebraic structures in complex analysis.

Contribution

It introduces a method to embed polydisk algebras into the disk algebra, allowing for the creation of subalgebras with arbitrary stable ranks.

Findings

Successful embedding of polydisk algebras into the disk algebra.

Construction of subalgebras with prescribed stable ranks.

Implications for the structure of uniform algebras.

Abstract

It is shown how to embed the polydisk algebras (finite and infinite ones) into the disk algebra $A(\overline{\mathbb D})$. As a consequence, one obtains uniform closed subalgebras of $A(\overline{\mathbb D})$ which have arbitrarily prescribed stable ranks