Uniform algebra

Summary

A uniform algebra is a closed subalgebra of the continuous complex functions on a compact Hausdorff space that contains the constants and separates points. It is a commutative Banach algebra with the sup norm. The main examples come from complex analysis, such as the disc algebra of functions holomorphic inside the disc and continuous up to the boundary. The Stone–Weierstrass theorem says the only uniform algebra on $X$ closed under complex conjugation is $C(X)$ itself. So the theory of uniform algebras studies the closed subalgebras that are not closed under conjugation.