In functional analysis, a uniform algebra A on a compact Hausdorff space X is a closed (with respect to the uniform norm) subalgebra of the C*-algebra C(X) (the continuous complex-valued functions on X) with the following properties:
the constant functions are contained in A for every x , y ∈ X {\displaystyle x,y\in X} there is f ∈ A {\displaystyle f\in A} with f ( x ) ≠ f ( y ) {\displaystyle f(x)\neq f(y)} . This is called separating the points of X. As a closed subalgebra of the commutative algebra (structure) Banach algebra C(X), a uniform algebra is itself a unital commutative Banach algebra (when equipped with the uniform norm). Hence, it is (by definition) a Banach function algebra. A uniform algebra A on X is said to be natural if the maximal ideals of A are precisely the ideals M x {\displaystyle M_{x}} of functions vanishing at a point x in X.
Abstract characterization If A is a unital commutative Banach algebra such that ‖ a 2 ‖ = ‖ a ‖ 2 {\displaystyle \|a^{2}\|=\|a\|^{2}} for all a in A, then there is a compact Hausdorff X such that A is isomorphic as a Banach algebra to a uniform algebra on X. This result follows from the spectral radius formula and the Gelfand representation.
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