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 ΦA with X, not just as sets but as topological spaces. The Gelfand representation is then an isomorphism
Read more about this topic: Gelfand Representation
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