Uniform algebra

Remarks

A uniform algebra is a commutative unital Banach algebra with $\norm{f^2} = \norm{f}^2$ for all $f$. Conversely, a commutative unital complex Banach algebra with $\norm{a^2} = \norm{a}^2$ for all $a$ is isometrically isomorphic, through the Gelfand transform, to a uniform algebra on its character space.

Example: the disc algebra $A(\mathbb{D})$ on the closed unit disc $X = \overline{\mathbb{D}}$, the continuous functions on $X$ that are holomorphic in the open disc. It is a subalgebra; it is closed because uniform limits of holomorphic functions are holomorphic; it contains the constants; and $f(z) = z$ separates points.

Non-example: on $X = [0, 1]$, let $A = \set{f \in C([0,1]) : f(0) = f(1)}$. It is a closed subalgebra containing the constants, but no $f \in A$ distinguishes $0$ from $1$, so clause 3 fails.

By the Stone–Weierstrass theorem, a uniform algebra on $X$ that is closed under complex conjugation equals $C(X)$.