Wolff’s Proof of the Corona Theorem

Abstract

The traditional definition of a function algebra is that it is any closed subalgebra of the continuous functions on a compact Hausdorff space. For us—at least at the beginning—the relevant compact Hausdorff space is the circle T. Classically, an important function algebra has been A(D)—the functions continuous on $\bar D$ and holomorphic on D. [We call A(D) the disk algebra.] Each such function can be identified with its restriction to the circle. And any such restriction has Fourier series with no coefficients of negative index. So this is clearly a subspace. It also follows by inspection that it is a subalgebra, and is closed.