Gelfand Representation - The C*-algebra Case

The C*-algebra Case

As motivation, consider the special case A = C₀(X). Given x in X, let be pointwise evaluation at x, i.e. . Then is a character on A, and it can be shown that all characters of A are of this form; a more precise analysis shows that we may identify Φ_A with X, not just as sets but as topological spaces. The Gelfand representation is then an isomorphism