Gelfand Representation - The Model Algebra

The Model Algebra

For any locally compact Hausdorff topological space X, the space C₀(X) of continuous complex-valued functions on X which vanish at infinity is in a natural way a commutative C*-algebra:

• The structure of algebra over the complex numbers is obtained by considering the pointwise operations of addition and multiplication.
• The involution is pointwise complex conjugation.
• The norm is the uniform norm on functions.

Note that A is unital if and only if X is compact, in which case C₀(X) is equal to C(X), the algebra of all continuous complex-valued functions on X.