A simple constructive proof of an analogue of the corona theorem

Abstract

We present a simple and constructive proof (i.e. a proof without using Gelfand theory) of an analogue of the Corona theorem for the Wiener algebras $W$ and ${W^ + }$ of absolutely convergent Fourier and Taylor series respectively, also the disc algebra $A(\overline D )$ and the subalgebras ${A^n}(\overline D )$ of functions whose $n$th derivatives extend continuously to $\overline D = \{ z:\left | z \right | \leqslant 1\}$.