Uniform algebra
A uniform algebra A on a compact Hausdorff topological 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 ∈{displaystyle in }
X there is f∈{displaystyle in }
f(y). This is called separating the points of X.
As a closed subalgebra of the commutative 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 precisely are the ideals Mx{displaystyle M_{x}}
Abstract characterization
If A is a unital commutative Banach algebra such that ||a2||=||a||2{displaystyle ||a^{2}||=||a||^{2}}
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